EXERCISE 1 (A) 9. The number of 9 digit numbers that can be formed by using the digits 1, 2, 3, 4 & 5 is : (A) 9 5 (B) 9!

ONLY ONE OPTION IS CORRECT 1. The number of arrangements which can be made using all the letters of the word LAUGH if the vowels are adjacent is (A) 10 (B) 24 (C) 48 (D) 120 2. The number of natural numbers from 1000 to 9999 (both inclusive) that do not have all 4 different digits is (A) 4048 (B) 4464 (C) 4518 (D) 4536 3. The number of different seven digit numbers that can be written using only three digits 1, 2 & 3 under the condition that the digit 2 occurs exactly twice in each number is : (A) 672 (B) 640 (C) 512 (D) none 4. Out of seven consonants and four vowels, the number of words of six letters, formed by taking four consonants and two vowels is (Assume that each ordered group of letter is a word): (A) 210 (B) 462 (C) 151200 (D) 332640 5. All possible three digits even numbers which can be formed with the condition that if 5 is one of the digit, then 7 is the next digit is : (A) 5 (B) 325 (C) 345 (D) 365 6. The number of six digit numbers that can be formed from the digits 1, 2, 3, 4, 5, 6 & 7 so that digits do not repeat and the terminal digits are even is : (A) 144 (B) 72 (C) 288 (D) 720 7. A new flag is to be designed with six vertical strips using some or all of the colour yellow, green, blue and red. Then, the number of ways this can be done such that no two adjacent strips have the same colour is (A) 12 81 (B) 16 192 (C) 20 125 (D) 24 216 8. A 5 digit number divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 & 5 without repetition. The total number of ways this can be done is : (A) 3125 (B) 600 (C) 240 (D) 216 9. The number of 9 digit numbers that can be formed by using the digits 1, 2, 3, 4 & 5 is : (A) 9 5 (B) 9! (C) 5 9 (D) 9 P 5 10. The number of arrangements of the letters ' a b c d ' in which neither a, b nor c, d come together is: (A) 6 (B) 12 (C) 16 (D) 8 11. The number of ways in which 5 different books can be distributed among 10 people if each person can get at most one book is : (A) 252 (B) 10 5 (C) 5 10 (D) 10 C 5.5! 12. The 9 horizontal and 9 vertical lines on an 8 8 chessboard form 'r' rectangles and 's' squares. The ratio s : r in its lowest terms is (A) 1 6 17 (B) 108 EXERCISE 1 (A) 4 (C) 27 (D) none Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 39

13. A student has to answer 10 out of 13 questions in an examination. The number of ways in which he can answer if he must answer atleast 3 of the first five questions is : (A) 276 (B) 267 (C) 80 (D) 1200 14. The number of three digit numbers having only two consecutive digits identical is (A) 153 (B) 162 (C) 180 (D) 161 15. The number of 6 - digit numbers that can be made with the digits 0, 1, 2, 3, 4, and 5 so that even digits occupy odd places, is (A)24 (B) 36 (C) 48 (D) none 16. The number of ways in which 6 men can be arranged in a row so that three particulars men are consective, is (A) 4 P 4 (B) 4 P 4 3 P 3 (C) 3 P 3 3 P 3 (D) none 17. The total number of words that can be made by writing the letters of the word PARAMETER so that no vowel is between the consonants is (A) 1440 (B) 1800 (C)2160 (D) none 18. The number of numbers of four different digits that can be fromed from the digits of the number 12 356 such that the numbers are divisible by 4, is (A) 36 (B) 48 (C) 12 (D) 24 19. The number of numbers of 9 different nonzero digits such that all the digits in the first four places are less than the central digit and all the digits in the last four places are greater than the central digit is : (A) 2(4!) (B) (4!) 2 (C) 8! (D) none 20. In the decimal system of numeration the number of 6 - digit number in which the digit in any place is greater than the digit to the left of it is (A) 210 (B) 84 (C) 126 (D) none 21. The number of 5 - digit numbers in which no two conseutive digits are identical is 2 3 4 (A) 9 8 (B) 9 8 (C) 9 5 (D) none 22. In the decimal system of numeration the number of 6 - digit numbers in which the sum of the digits is divisible by 5 is (A) 180000 (B) 540000 (C) 5 10 5 (D) none 23. Two teams are to play a series of 5 matches between them. A match ends in a win or lass or draw for a team. A number of people forecast the result of each match and no two people make the same forecast for the series of matches. The smallest group of people in which one person forecasts correctly for all the matches will contain n people, when n is (A) 81 (B) 243 (C) 486 (D) none 24. Total number of 6 - digit numbers in which all the odd digits and only odd digits appear, is (A) 5 (6!) 2 1 (B) 6! (C) (6!) 2 (D) none 25. Four couples ( husband and wife ) decide to form a commitee of four members. The number of different that can be formed in which no couple find a place is (A) 10 (B) 12 (C) 14 (D) 16 Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 40

26. The number of ways to fill each of the four cells of the table with a distinct nautral number such that the sum of the numbers is 10 and the sums of the numbers placed diagonally are equal, is (A) 2! 2! (B) 4! (C) 2(4!) (D) none of these 27. The interior angles of a regular polygon measure 150º each. The number of diagonals of the polygon is (A) 35 (B) 44 (C) 54 (D) 78 28. Number of different natural numbers which are smaller than two hundred million & using only the digits 1 or 2 is (A) (3). 28-2 (B) (3). 28-1 (C) 2 (2 9-1) (D) none 29. The number of n digit numbers which consists of the digits 1 & 2 only if each digit is to be used atleast once, is equal to 510 then n is equal to: (A) 7 (B) 8 (C) 9 (D) 10 30. 18 points are indicated on the perimeter of a triangle ABC (see figure). How many triangles are there with vertices at these points? (A) 331 (B) 408 (C) 710 (D) 711 31. A question paper on mathematics consists of twelve questions divided into three parts A, B and C, each containing four questions. In how many ways can an examinee answer five questions, selecting atleast one from each part. (A) 624 (B) 208 (C) 2304 (D) none 32. If m denotes the number of 5 digit numbers if each successive digits are in their descending order of magnitude and n is the corresponding figure. When the digits and in their ascending order of magnitude then (m n) has the value (A) 10 C 4 (B) 9 C 5 (C) 10 C 3 (D) 9 C 3 33. There are m points on a straight line AB & n points on the line AC none of them being the point A. Triangles are formed with these points as vertices, when (i) A is excluded (ii) A is included. The ratio of number of triangles in the two cases is: (A) m n 2 m n m n 2 (B) m n 1 (C) m n 2 m n 2 (D) m (n 1) (m 1) (n 1) 34. Number of ways in which 9 different prizes be given to 5 students if one particular boy receives 4 prizes and the rest of the students can get any numbers of prizes, is : (A) 9 C 4. 2 10 (B) 9 C 5. 5 4 (C) 4. 4 5 (D) none 35. There are n persons and m monkeys (m > n). Number of ways in which each person may become the owner of one monkey is (A) n m (B) m n (C) m P n (D) mn Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 41

36. Let there be 9 fixed points on the circumference of a circle. Each of these points is joined to every one of the remaining 8 points by a straight line and the points are so positioned on the circumference that atmost 2 straight lines meet in any interior point of the circle. The number of such interior intersection points is (A) 126 (B) 351 (C) 756 (D) none of these 37. The number of 5 digit numbers such that the sum of their digits is even is : (A) 50000 (B) 45000 (C) 60000 (D) none 38. The number of ways in which 8 non-identical apples can be distributed among 3 boys such that every boy should get atleast 1 apple & atmost 4 apples is K 7 P 3 where K has the value equal to : (A) 88 (B) 66 (C) 44 (D) 22 39. A rack has 5 different pairs of shoes. The number of ways in which 4 shoes can be chosen from it so that there will be no complete pair is (A) 1920 (B) 200 (C) 110 (D) 80 40. The greatest possible number of points of intersection of 9 different straight lines & 9 different circles in a plane is: (A) 117 (B) 153 (C) 270 (D) none Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 42

EXERCISE 1 (B) MORE THAN ONE OPTIONS MAY BE CORRECT 1. 2n P n is equal to : n 2n (A) ( n 1)( n 2)...(2 n) (B) 2 [1 3 5...(2n 1)] (C) (2) (6) (10)...(4 n 2) (D) n!( C n ) 2. If 100! 2 3 5 7..., then : 1 (A) 97 (B) ( 1) (C) 2 1 2 (D) 1 3 3. A student has to answer 10 out of 13 questions in an examination. The number of ways in which he can answer if he must answer atleast 3 of the first five questions is : (A) 276 (B) 267 (C) 13 C 5 10 C3 (D) 5 C 8 C 5 C 8 C 5 C 8 C 3 7 4 6 5 5 4. If k is odd then k C r is maximum for r equal to : (A) 1 ( 1) 2 k (B) 1 ( 1) 2 k (C) k 1 (D) k 5. The number of non-negative integral solutions of x1 x2 x3 x4 n (where n is a positive integer) is : n 3 (A) C n 4 (B) C n 5 (C) C n 4 (D) 3 4 6. You are given 8 balls of different colour (black, white,...). The number of ways in which these balls can be arranged in a row so that the two balls of particular colour (say red & white) may never come together is : (A) 8! 2. 7! (B) 6. 7! (C) Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 43 5 C n 7 2 6! C 2 (D) none 7. If S 0! 1 1! 2 2!... n n!, then : (A) n S (B) ( n 1) S (C) n! S (D) ( n 1)! S 8. Letters of the word SUDESH can be arranged in: (A) 120 ways when two vowels are together (B) 180 ways when vowels occur in alphabetical order (C) 24 ways when vowels and consonants occupy their respective place (D) 240 ways when vowels do not occur together 9. The number of ways in which 10 candidates A1, A2,..., A 10 can be ranked so that A 1 is always above A 2 is : (A) 10! 2 (B) 7 4 2 1 2 3 5 7 (C) 8 2 2 5 (D) 4 2 3 5 7 10. There are 10 seats in the first row of a theater of which 4 are to be occupied. The number of ways of arranging 4 persons so that no two persons sit side by side is (A) 7 C 4 (B) 7 4 P 3 (C) 7 4! C 3 (D) 840 11. On Diwali, all the students of a class send greeting cards to one another. If the postmen deliver 1640 greeting cards to the students of this class,then the number of students in the class is : (A) 40 (B) 41 (C) A prime (D) An even number

12. Let x be the number of 5 digit numbers sum of whose digits is even and y be the number of 5 digit numbers sum of whose digits is odd, then: (A) x y (B) x y 90,000 (C) x 45,000 (D) x y 13. A man is dealt a poker hand (consisting of 5 cards) from an ordinary pack of 52 playing cards. The number of ways in which he can be dealt a straight (a straight is five consecutive values not of the same suit, eg. {Ace 2. 3. 4. 5}, (2, 3, 4, 5, 6}... & {10. J. Q. K. Ace} is (A) 10 (4 5 4) (B) 4! 2 10 (C) 10. 2 10 (D) 10200 14. The sides AB, BC and CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these interior points as vertices is : (A) 60 (B) 205 (C) 115 (D) 405 15. An ice cream parlour has ice creams in eight different varieties. Number of ways of chossing 3 ice creams taking atleast two ice creams of the same variety, is : (A) 56 (B) 64 (C) 10 C 8 C (D) 10 C 8 C 2 2 3 3 16. A seven digit number is divisible by 3 is to be formed using 7 out of numbers 1,2,3,...,9. The number of ways in which this can be done is : (A) 12(7!) (B) 7(8!) (C) 2(8!) (D) None of these 17. The number of ways in which 10 students can be divided into three teams, one containing 4 and others 3 each, is 10! (A) (B) 2100 (C) 4! 3! 3! 10 C 5 C 4 3 18. The number 1000 C 500 is divisible by : (A) 7 (B) 13 (C) 191 (D) 201 (D) 10! 1. 6! 3! 3! 2 19. Identify the correct statements (s) (A) Number of zeroes standing at the end of 125! is 30 (B) A telegraph has 10 arms and each arm is capable of 9 distinct positions excluding the position of rest.the number of signals that can be transmitted is 10 10 1 (C) Number of numbers greater than 4 lacs which can be formed by using only the digits 0 2 2 4 4 and 5 is 90 (D) In a table tennis tournamant every player plays with every other player. If the number of games played is 5050 then the number of players in the tournament is 100 20 20. Let N (20)! and M C 10 (A) The number of zeros at the end of N is 4 (B) the last digit of N /10 is 8 (C) Exponent of 5 in M is 0 (D) Exponent of 7 in M is 0 Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 44

COMPREHENSION TYPE PASSAGE - 1 Consider the 10 letter-word W = CURRICULUM. 21. Number of ways in which 5 lettered words can be formed using the letters from the word W if each 5 lettered word has exactly 3 different letters, is (A) 360 (B) 560 (C) 610 (D) 720 22.. Number of ways in which all the letters of the word W can be arranged if vowels are to be separated is k (7!), then the value of k is equal to (A) 5 (B) 4 (C) 3 (D) 2 23. Number of ways in which all the letters of the word W can be arranged if relative order of vowels and consonants do not change is equal to (A)360 (B) 720 (C) 1080 (D) 1440 PASSAGE - 2 Consider the set S = {0, 1, 2,..., 9}. 24. Number of all five digit numbers that can be formed using the digits from S if their digits are in due order, is (A) 126 (B) 252 (C) 310 (D) 378 25. Number of all five digit numbers that can be formed using the digits from S containing 2 alike and 3 other alike digit, is (A) 810 (B) 750 (C) 720 (D) None 26. Number of 10-digit prime numbers that can be formed using each and every digit of S, is (A) 0 (B) 1 (C) 10 (D) 100 PASSAGE - 3 If n( A ) denotes the number of elements in the finite set A and A1, A 2,... are finite sets, then : n n n Ai n( Ai ) n( Ai Aj ) i 1 i 1 i j n n 1 n( Ai Aj Ak )... ( 1) n Ai i j k i 1 27. Five letters are put in 5 addressed envelops randomly, the number of ways in which no letters goes to the correct envelop is : (A) 41 (B) 43 (C) 44 (D) 46 28. The number of postive integers 100 which are not divisible by 2, 3 or 5 is : (A) 25 (B) 26 (C) 29 (D) 30 29. The number of integral solutions of x1 x2 x3 x4 30 such that 0 x i 10, is : (A) 84 (B) 87 (C) 90 (D) None Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 45

PASSAGE - 4 If p is a prime. then exponent of p in n! equals p ( ) n n n E n... 2 3 p p p 30. The exponent of 7 in 1000 C 500 is : (A) 0 (B) 1 (C) 2 (D) 3 31. The number of zeros at the 50! (A) 11 (B) 10 (C) 12 (D) 9 32. The largest two digit prime that divides 200 C 100 is : (A) 59 (B) 53 (C) 47 (D) None of these ASSERTION REASONING TYPE (A) Stat.-1 is true, Stat.-2 is true and Stat.-2 is correct explanation for Stat.-1. (B) Stat.-1 is true, Stat.-2 is true but Stat.-2 is NOT the correct explanation for Stat.-1. (C) Statement-1 is true, Statement-2 is false. (D) Statement-1 is false, Statement-2 is true. 33. Statement-1 : The product of three consecutive natural numbers is divisible by 6. Statement-2 : The product of n consecutive natural numbers is divisible by n! 34. Statement-1 : The expression 40 60 40 60... r 0 r 1 1 attains maximum value when r 50. 2n Statement-2 : is maximum when r. r n 35. Statement-1 : The number of non-negative integral solutions of x1 x2... x20 100 is 120. 20 Statement-2 : The number of ways of distributing n identical objects among r persons 36. Let n, m N and n m. Statement-1 : giving zero or more objects to a person is n r 1. r 1 n n 1 n 2 m n 2 2 3... (n m 1) m m m m m 1 Statement-2 : n n 1 n 2 m... m m m m n 1 m 1 37. Statement-1 : Then number of onto function f from A {1, 2, 3, 4, 5, 6} to B {7, 8, 9} such that f ( i) f ( j) i j is 6. Statement-2 : The number of permutations of 7, 8, 9 taken 3 at a time is 6. Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 46

MATRIX MATCH TYPE 38. Consider 6 children C 1, C 2, C 3, C 4, C 5 and C 6 of different heights, child with higher suffix is taller. Column I Column II (A) Number of ways in which these six children can line up (P) 20 in a single row so that none of them is standing between the two children taller than him, is (B) Number of ways in which they appear in a line if 3 children (Q) 32 C 1, C 2 and C 3 are in ascending order of their height (not necessarily successive) is, (C) Number of ways in which 10 alike marbles can be distributed (R) 90 among them if every child gets atleast one marble but not more than 4, is (D) Number of ways in which they can be stand in a rectangular (S) 120 array of 3 rows and two columns so that any child in a column is taller than the child in the same column immediately in front of him, is e.g. Column I Column II C C C 3 2 1 C C C 6 5 4 39. Suppose a set A consists of 10 distinct elements x 1, x 2, x 3,..., x 10 The number of subsets of A which contain Column I Column II (A) None of x1, x2, x 3 (P) 512 (B) Each of x1, x2, x 3 (Q) 896 (C) At least one of x1, x2, x 3 (R) 128 (D) At most one of x1, x2, x 3 (S) 256 40. Consider all possible permutaions of the letters of the word ENDEANOEL. Column I Column II (A) The number of permutaitons containing the word ENDEA is (P) 5! (B) (C) (D) The number of permutaions in which the letter E occurs in the first and the last positions is (Q) 2 5! The number of permutaions in which none of the letters D, L, N occurs in the last five positions is (R) 7 5! The number of permutaions in which the letters A, E, O occur only in odd position is (S) 21 5! Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 47

INTEGER TYPE EXERCISE 1 (C) 1. Number of ways in which 7 people can occupy six seats, 3 seats on each side in a first class railway compartment if two specified persons are to be always included and occupy adjacent seats on the same side, is (5!) k then k has the value equal to. 2. Let P n denotes the number of ways in which three people can be selected out of ' n ' people sitting in a row, if no two of them are consecutive. If, P n + 1 P n = 15 then the value of ' n ' is. 3. Number of three digit number with atleast one 3 and at least one 2 is. 4. Total number of ways in which 6 + & 4 signs can be arranged in a line such that no 2 signs occur together is. 5. Define a 'good word' as a sequence of letters that consists only of the letters A, B and C and in which A never immidiately followed by B, B is never immediately followed by C, and C is never immediately followed by A. If the number of n-letter good words are 384, then the value of n is. 6. Fifty college teachers are surveyed as to their possession of colour TV, VCR and tape recorder. Of them, 22 own colour TV, 15 own VCR and 14 own tape recorders. Nine of these college teachers own exactly two items out of colour TV, VCR and tape recorders ; and, one college teacher owns all three. how many of the 50 college teachers own none of three, colour TV, VCR or tape recorder? 7. A road network as shown in the figure connect four cities. In how many ways can you start from any city (say A) and come back to it without travelling on the same road more than once? 8. There are 6 boxes numbered 1, 2,... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutive. The total number of ways in which this can be done, is. 1000! 9. The least value of the positive integer n, for which n 13 is not an integer, is. 10. Consider the lines x = k and y = k, k {1, 2,..., 9}. The number of non-congruent rectangles, whose sides are along these lines, is. 11. The number of representations of the number 7056, as a product of 2 factors, is. 12. The number of 3 element subsets of {1, 2,...,n}, in which the least element is 3 or the greatest element is 7, is 33. The value of n is. 13. A point, P, is at a distance of 12 cm from the centre of a circle of radius 13cm. The number of chords of the circle passing through P, and which have integral lengths, is. P Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 48

14. The expression 20 C 0. 20 C 1 + 20 C 1. 20 C 2 +... + 20 C 19 20 C 20 equals n C r, where r > [ 2 n ]. The value of r is {where [ ] is the step function}. 15. One hundred management students who read at least one of the three business magazines are surveyed to study the readership pattern. It is found that 80 read Business India, 50 read Business world, and 30 read Business Today. Five students read all the three magazines. How many read exactly two magazines? 16. Two classrooms A and B having capacity of 25 and (n 25) seats respectively.a n denotes the number of possible seating arrangements of room 'A', when 'n' students are to be seated in these rooms, starting from room 'A' which is to be filled up full to its capacity. If A n A n 1 = 25! ( 49 C 25 ) then the value of 'n' is. 17. The number of ways in which the number 108900 can be resolved as a product of two factors is. 18. In maths paper there is a question on "Match the column" in which column A contains 6 entries & each entry of column A corresponds to exactly one of the 6 entries given in column B written randomly. 2 marks are awarded for each correct matching & 1 mark is deducted from each incorrect matching. A student having no subjective knowledge decides to match all the 6 entries randomly. The number of ways in which he can answer, to get atleast 25 % marks in this question is. 19. There is an alphabet of k letters, k 5. The total number of palindromes, each of 5 letters, that can be formed, is 729. The value of k is. 20. In a chess tournament, each participant was supposed to play exactly one game with each of the others. However, two participants withdraw after having played exactly 3 games each, but not with each other. The total number of games played in the tournament was 84. How many participants were there in all? Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 49

ONLY ONE OPTION IS CORRECT 1. In the figure, two 4 - digit numbers are to be formed by filling the place with digits. The number of different ways in which the places can be filled by digits so that the sum of the numbers formed is also a 4 - digit number and in no place the addition in with carrying, is Th H T U EXERCISE 2 (A) (A) 55 4 (B) 220 (C) 45 4 (D) none 2. 5 Indian & 5 American couples meet at a party & shake hands. If no wife shakes hands with her own husband & no Indian wife shakes hands with a male, then the number of hand shakes that takes place in the party is (A) 95 (B) 110 (C) 135 (D) 150 3. An English school and a Vernacular school are both under one superintendent. Suppose that the superintendentship, the four teachership of English and Vernacular school each, are vacant, if there be altogether 11 candidates for the appointments, 3 of whom apply exclusively for the superintendentship and 2 exclusively for the appointment in the English school, the number of ways in which the different appointments can be disposed of is (A) 4320 (B) 268 (C) 1080 (D) 25920 4. A committee of 5 is to be chosen from a group of 9 people. Number of ways in which it can be formed if two particular persons either serve together or not at all and two other particular persons refuse to serve with each other, is (A) 41 (B) 36 (C) 47 (D) 76 5. Seven different coins are to be divided amongst three persons. If no two of the persons receive the same number of coins but each receives atleast one coin & none is left over, then the number of ways in which the division may be made is : (A) 420 (B) 630 (C) 710 (D) none 6. An old man while dialing a 7 digit telephone number remembers that the first four digits consists of one 1's, one 2's and two 3's. He also remembers that the fifth digit is either a 4 or 5 while has no memorising of the sixth digit, he remembers that the seventh digit is 9 minus the sixth digit. Maximum number of distinct trials he has to try to make sure that he dials the correct telephone number, is (A) 360 (B) 240 (C) 216 (D) none 7. Number of ways in which 9 different toys be distributed among 4 children belonging to different age groups in such a way that distribution among the 3 elder children is even and the youngest one is to receive one toy more, is : (A) 2 (5!) 8 (B) 9! 2 9! (C) 3 3!(2!) (D) none 8. In an election three districts are to be canvassed by 2, 3 & 5 men respectively. If 10 men volunteer, the number of ways they can be alloted to the different districts is : (A) 10! 2! 3! 5! (B) 10! 2! 5! Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 50 (C) 10! 2 ( 2!) 5! 10! (D) 2 (2!) 3! 5!

9. There are 10 red balls of different shades & 9 green balls of identical shades. Then the number of arranging them in a row so that no two green balls are together is : (A) (10!). 11 P 9 (B) (10!). 11 C 9 (C) 10! (D) 10! 9! 10. A shelf contains 20 different books of which 4 are in single volume and the others form sets of 8, 5 and 3 volumes respectively. Number of ways in which the books may be arranged on the shelf, if the volumes of each set are together and in their due order is (A) 20! 8! 5! 3! (B) 7! (C) 8! (D) 7. 8! 11. If all the letters of the word "QUEUE" are arranged in all possible manner as they are in a dictionary, then the rank of the word QUEUE is : (A) 15 th (B) 16 th (C) 17 th (D) 18 th 12. There are 12 different marbles to be divided between two children in the ratio 1 : 2. The number of ways it can be done is : (A) 990 (B) 495 (C) 600 (D) none 13. All the five digits number in which each successive digit exceeds its predecessor are arranged in the increasing order of their magnitude. The 97 th number in the list does not contain the digit (A) 4 (B) 5 (C) 7 (D) 8 14. The number of different ways in which five 'dashes' and eight 'dots' can be arranged, using only seven of these 13 'dashes' & 'dots' is : (A) 1287 (B) 119 (C) 120 (D) 1235520 15. There are n identical red balls & m identical green balls. The number of different linear arrangements consisting of "n red balls but not necessarily all the green balls" is x C y then (A) x = m + n, y = m (B) x = m + n + 1, y = m (C) x = m + n + 1, y = m + 1 (D) x = m + n, y = n 16. Number of different words that can be formed using all the letters of the word "DEEPMALA" if two vowels are together and the other two are also together but separated from the first two is (A) 960 (B) 1200 (C) 2160 (D) 1440 17. The number of ways in which 10 boys can take positions about a round table if two particular boys must not be seated side by side is : (A) 10 (9)! (B) 9 (8)! (C) 7 (8)! (D) none 18. In a unique hockey series between India & Pakistan, they decide to play on till a team wins 5 matches. The number of ways in which the series can be won by India, if no match ends in a draw is : (A) 126 (B) 252 (C) 225 (D) none 19. Number of ways in which n things of which r alike & the rest different can be arranged in a circle distinguishing between clockwise and anticlockwise arrangement, is : (A) (C) (n r 1)! r! (n 1)! (r 1)! Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 51 (B) (D) (n 1)! r ( n 1)! r!

20. A gentleman invites a party of m + n (m n) friends to a dinner & places m at one table T 1 and n at another table T 2, the table being round. If not all people shall have the same neighbour in any two arrangement, then the number of ways in which he can arrange the guests, is (A) ( m n)! 4 mn (B) 1 2 ( m n)! mn (C) 2 (m n)! mn (D) none 21. Delegates from 9 countries includes countries A, B, C, D are to be seated in a row. The number of possible seating arrangements, when the delegates of the countries A and B are to be seated next to each other and the delegates of the countries C and D are not to be seated next to each other is : (A) 10080 (B) 5040 (C) 3360 (D) 60480 22. There are 12 guests at a dinner party. Supposing that the master and mistress of the house have fixed seats opposite one another, and that there are two specified guests who must always, be placed next to one another ; the number of ways in which the company can be placed, is: (A) 20. 10! (B) 22. 10! (C) 44. 10! (D) none 23. Let P n denotes the number of ways of selecting 3 people out of 'n' sitting in a row, if no two of them are consecutive and Q n is the corresponding figure when they are in a circle. If P n Q n = 6, then 'n' is equal to (A) 7 (B) 8 (C) 9 (D) 10 24. There are (p + q) different books on different topics in Mathematics. (p q) If L = The number of ways in which these books are distributed between two students X and Y such that X get p books and Y gets q books. M = The number of ways in which these books are distributed between two students X and Y such that one of them gets p books and another gets q books. N = The number of ways in which these books are divided into two groups of p books and q books then, (A) L = M = N (B) L = 2M = 2N (C) 2L = M = 2N (D) L = M = 2N 25. Let m denote the number of ways in which 4 different books are distributed among 10 persons, each receiving none or one only and let n denote the number of ways of distribution if the books are all alike. Then (A) m = 4n (B) n = 4m (C) m = 24n (D) none 26. The number of ways in which we can arrange n ladies & n gentlemen at a round table so that 2 ladies or 2 gentlemen may not sit next to one another is : (A) (n - 1)! (n - 2)! (B) (n!) (n - 1)! (C) (n + 1)! (n)! (D) none 27. The number of ways in which 10 identical apples can be distributed among 6 children so that each child receives atleast one apple is : (A) 126 (B) 252 (C) 378 (D) none of these 28. The number of all possible selections of one or more questions from 10 given questions, each equestion having an alternative is : (A) 3 10 (B) 2 10 1 (C) 3 10 1 (D) 2 10 29. Six persons A, B, C, D, E and F are to be seated at a circular table. The number of ways this can be done if A must have either B or C on his right and B must have either C or D on his right is : (A) 36 (B) 12 (C) 24 (D) 18 Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 52

30. The number of ways of choosing a committee of 2 women & 3 men from 5 women & 6 men, if Mr. A refuses to serve on the committee if Mr. B is a member & Mr. B can only serve, if Miss C is the member of the committee, is : (A) 60 (B) 84 (C) 124 (D) none 31. There are 2 identical white balls, 3 identical red balls and 4 green balls of different shades. The number of ways in which they can be arranged in a row so that atleast one ball is separated from the balls of the same colour, is : (A) 6 (7! - 4!) (B) 7 (6! - 4!) (C) 8! - 5! (D) none 32. Sameer has to make a telephone call to his friend Harish, Unfortunately he does not remember the 7 digit phone number. But he remembers that the first three digits are 635 or 674, the number is odd and there is exactly one 9 in the number. The maximum number of trials that Sameer has to make to be successful is (A) 10,000 (B) 3402 (C) 3200 (D) 5000 33. Six people are going to sit in a row on a bench. A and B are adjacent. C does not want to sit adjacent to D. E and F can sit anywhere. Number of ways in which these six people can be seated, is (A) 200 (B) 144 (C) 120 (D) 56 34. Boxes numbered 1, 2, 3, 4 and 5 are kept in a row, and they are necessarily to be filled with either a red or a blue ball, such that no two adjacent boxes can be filled with blue balls. Then how many different arrangements are possible, given that the balls of a given colour are exactly identical in all respects? (A) 8 (B) 10 (C) 13 (D) 22 35. The combinatorial coefficient C(n, r) can not be equal to the (A) number of possible subsets of r members from a set of n distinct members. (B) number of possible binary messages of length n with exactly r 1's. (C) number of non decreasing 2-D paths from the lattice point (0, 0) to (r, n). (D) number of ways of selecting r things out of n different things when a particular thing is always included plus the number of ways of selecting 'r' things out of n, when a particular thing is always excluded. 36. Given 11 points, of which 5 lie on one circle, other than these 5, no 4 lie on one circle. Then the maximum number of circles that can be drawn so that each contains atleast three of the given points is: (A) 216 (B) 156 (C) 172 (D) none 37. The number of ways of dividing a pack of 52 cards into 4 equal groups is 52! (A) 13! 4 (B) 52 C 4 (C) 52! 4! 13! 4 (D) none of these 38. The number of non-negative integral solutions of 3x +y + z = 24 is (A) 117 (B) 108 (C) 225 (D) none of these 39. Eight people, including A and B, are to be seated around two identical tables, each having a capacity of 4. The number of seating arrangements, so that A and B are not at the same table, is (A) 360 (B) 1440 (C) 720 (D) 2880 Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 53

40. Observe that 3 3 + 4 3 + 5 3 = 6 3. The number of cubes of positive integers, which can be expressed as a sum of three integer cubes, is (A) 1 (B) 3 (C) 6 (D) infinitely many 41. The number of zeroes, at the end of 50!, is (A) 5 (B) 10 (C) 11 (D) 12 42. The number of unordered pairs (A,B), of subsets of the set S ={1, 2,..., 10}, such that A B = and A B = S, is (A) 2 10 (B) 2 9 (C) 10 C 2 (D) 1+2 5 43. A positive integer n is called strictly ascending if its digits are in the increasing order. e.g. 2368 and 147 are strictly ascending but 43679 is not. The number of strictly ascending numbers < 10 9 is (A) 2 9 1 (B) 2 9 (C) 2 9 +1 (D) 9! 44. How many nine digit numbers can be formed from the number 223355888 by rearranging its digits, so that the odd digits occupy the even positions? (A) 16 (B) 36 (C) 60 (D) 180 45. Ten people, including A and B, are to be seated at a rectangular table having 5 seats on each of its longer sides. The number of seating arrangements, so that A and B are neither next to each other nor directly opposite each other, is (A) 9! 2 (B) 8! (C) 8! 28 (D) 8! 64 46. m, n are positive integers such that gcd (m, n) = 1 and mn= 25!. The number of rational numbers m < 1 is n (A) 256 (B) 512 (C) 108 (D) 64 47. a, b, c {1, 2,... 14}. Let P(x) = ax 2 + 2bx + c. What is the number of polynomials P(x) such that x +1 divides P(x)? (a, b, c are distinct) (A) 14 C 3 (B) 7 C 2. 7 C 2 (C) 7 C 2 (D) 2( 7 C 2 + 7 C 2 ) 48. The number of positive integers of upto 9 digits, in which the digit 1 appears atleast once, is (A) 8 9 (B) 10 9 9 9 (C) 8 9 1 (D) 10 9 1 49. The number of ways of choosing a white square and a black square, on a chess-board, so that they do not belong to the same row or column,is (A) 32 2 (B) 32 24 (C) 32 25 (D) none of these 50. The number of arrangements of all the cards in a deck, such that the red and the black cards alternate, is (A) (26!) 2 (B) 2(26!) 2 (C) 2(52!) 2 (D) (52!) 2 51. n, r are positive integers n > r > 5. If 13 C 3 + 2. 13 C 4 + 13 C 5 = n C r, then r equals (A) 10 (B) 8 (C) 9 (D) 11 52. Four ordinary dice are tossed. The number of outcomes, in which all the dice show the same number, is (A) 4 6! (B) 4 6 4 (C) 6 (D) 4 6 Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 54

53. Eight people are to be seated, including A and B, four at a round table and four at a straight table. The number of arrangements, in which A and B are not at the round table, is (A) 7920 (B) 2160 (C) 10,080 (D) none of these 54. The number of arrangements of all the cards in a complete deck, such that cards of the same suit are together, is (A) 52! 4! 13! (B) 4! 4 52! (C) (13!) 4.4! (D) 13! 4 55. The number of teams for a mixed doubles tennis match, that can be formed from eight couples, is (A) 784 (B) 1820 (C) 910 (D) 1568 56. Eight players take part in a tournament where each player competes with some other player. The number of pairings is (A) 28 (B) 56 (C) 64 (D) 105 57. Ten parabolae are drawn in a plane. Any two parabolae intersect in two real, and distinct, points. No three parabolae are concurrent.the total number of disjoint regions of the plane is (A) 105 (B) 99 (C) 101 (D) 103 58. There are m men and n women. k distinguishable objects are to be distributed among them. (Each person can receive any number of objects.) The number of distributions, such that no woman receives any object, is (A) m k (B) (m+n) k m k (C) (m n) k (D) none of these 59. Consider the letters of the word EQUATIONS. What is the number of arrangements of the letters in this word so that the vowels appear, not necessarily successively, in the dictionary order? (A) 9! 5!4! (B) 9! 4! (C) 9! (D) 9! 5! 60. The adjoining diagram shows 8 clay targets, arranged in 3 columns, to be shot by 8 bullets. The number of ways in which they can be shot, such that no target is shot before all the targets below it, if any, are first shot,is (A) 560 (B) 72 (C) 40320 (D) none of these 61. The number of lattice points in the 1 st quadrant, lying on the straight line 3x + 5y = 283, is (A) 21 (B) 20 (C) 18 (D) 19 B 62. A and B are finite sets. There are 2 elements in A B and 21 elements in A B. Hence the number of elements in A B ( ( A B ) ( A B) ) is (A) 6 (B) 10 (C) 8 (D) 7 63. Eight identical rooks are to be placed on an 8 8 chess-board. The number of ways of doing this, so that no two rooks are in attacking positions, is (A) 8 (B) 8! (C) 8 8 (D) 8 2 Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 55 A

64. The maximum number of intersection points of n circles and n straight lines, among them selves, is 80. The value of n is (A) 7 (B)6 (C) 5 (D) 8 65. 900 n digit number, using only the digits 2, 5 and 7, are to be formed. The least value of n is (A) 3 (B)7 (C) 8 (D) 5 66. 12 normal dice are thrown once. The number of ways in which each of the values 2, 3, 4, 5 and 6 occurs exactly twice is: [ 1,1, 2,2, 3,3, 4,4, 5,5, 6,6 can come in any order ] (A) ( 12)! 6 (B) ( 12)! (12)! 2 6. 6! (C) 2 6 (D) none 67. An alphabet contains a A S and b B S. (In all a + b letters). The number of words, each containing all the A S and any number of B S, is (A) a + b C b (B) a + b + 1 C a (C) a + b + 1 C b (D) none of these 68. A = {1, 11, 21, 31,..., 541, 551}. B is a subset of A such that x +y 552, for any x, y B. The maximum number of elements in B is (A) 26 (B) 30 (C) 29 (D) 28 69. The number of ways of arranging the letters AAAAA, BBB, CCC, D, EE & F in a row if the letter C are separated from one another is : 12! (A) 13 C 3 5! 3! 2! (B) 13! 5! 3! 3! 2! (C) 14! 3! 3! 2! (D) none 70. The maximum number of different permutations of 4 letters of the word "EARTHQUAKE" is : (A) 2910 (B) 2550 (C) 2190 (D) 2091 Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 56

SUBJECTIVE TYPE EXERCISE 2 (B) 1. In how many ways can 2n people be seated, n at a round table and n in a row? 2. Find the number of selections of 5 cards from a pack such that all 4 suits are present. 3. There are 17 people including A and B. In how many ways can be 12 of them be chosen if both A and B are not chosen together? 4. In how many ways can 3 distinct numbers be chosen from amongst 1 to 30 so that their sum is even? 5. In how many ways can 27 distinct books be distributed among A, B, C so that C gets half as much as A and B together? 6. 20 lines are drawn in a plane. No two lines are parallel and no three are concurrent. Show that they divide the plane into 211 disjoint parts. 7. In how many ways can a 12 step staircase be climbed taking 1 step or 2 steps at a time? 8. Each side of an equilateral ABC is divided into 6 equal parts. The corresponding points of subdivison are joined. Find the number of equilateral triangles oriented the same way as ABC. 9. Let n = 10 6. Evaluate d n log d. 10 10. How many arrangements of the 9 letters a, b, c, p, q, r, x, y, z are there such that y is between x and z? (Any two, or all three, of the letters x, y, z, may not be consecutive.) 11. From amongst 8 married couples, a team is to be selected for a mixed doubles match. Find the total number of teams in which spouses are not included. 12. Let n = 180. Find the number of positive integral divisors of n 2, which do not divide n. 13. How many arrangements of the letters of MISSISSIPPI have no consecutive S s? 14. Find the number of ways of keeping 2 identical kings on an 8 8 chess-board so that they are not in adjacent squares. (Two squares are adjacent when they have a common side.) 15. A coin is tossed 10 times. Find the number of outcomes in which 2 heads are not successive. 16. There are 11 seats in a row. Five people are to be seated. Find the number of seating arrangements, if i ) the central seat is to be kept vacant ; and ii) for every pair of seats symmetric with respect to the central seat, one seat is vacant. 17. Take a convex octagon in which no two diagonals are parallel and no three are concurrent inside the polygon. Find the number of intersection points, lying inside the polygon, of the diagonals. 18. How many hexagons can be constructed by joining the vertices of a quindecagon (15 sides) if none of the sides of the hexagon is also the side of the 15-gon. Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 57

19. There are five distinguishable pairs of gloves to be given to 5 persons. Each person must get a left glove and a right glove. Find the number of distributions so that no person gets a proper pair. 20. Six distinguishable marbles are to be distributed into 3 distinguishable boxes. Find the number of distributions so that no box is empty. 21. Find the number of (m + n) - digit sequences with m 0 S and n 1 S such that no two 1 S are adjacent, n m +1. 22. Find the number of ways to pave a 1 7 rectangle by 1 1, 1 2, 1 3 tiles, if tiles of the same size are indistinguishable. 23. Find the number of seating arrangements of 6 persons at three identical round tables if every table must be occupied. 24. All the 5 digit numbers, formed by permuting the digits 1, 2, 3, 4 and 5 are arranged in the increasing order. Find : - i) the rank of 35421 ii) the 100 th number. 25. Show that the number of positive integral divisors of 111... 1(2010 times) is even. 26. Let n = 2 6. 3 4. 5 2.7 4. Find the number of positive integral divisors of n which are greater than n. 27. Find the number of ordered pairs (x, y) of positive integers such that 1 1 1. x y 100 28. How many increasing 3 term arithmetic progressions can be formed whose terms are from amongst 1, 2, 3,..., 100? 29. How many unordered pairs {a, b} of positive integers a and b are there such that lcm (a, b)= 1,26,000? (Note : An unordered pair {a, b} means {a, b} = {b, a}) 30. P is a 21 sided regular polygon. There are exactly 21 C 3 = 1330 triangles whose vertices are vertices of P. How many of these triangles are :- (i) acute? (ii) isosceles? (Isosceles include equilateral.) Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 58

ONLY ONE OPTION IS CORRECT EXERCISE 3 (A) 1. No. of different garlands using 3 flowers of one kind and 12 flowers of seconds kind is (A) 19 (B) 11! 2! (C) 14 C 2 (D) none of these 2. Number of even divisiors of 504 is (A) 12 (B) 24 (C) 6 (D) 18 3. Let p be a prime number such that p 3. Let n = p! + 1. The number of primes in the list n + 1, n + 2, n + 3,... n + p 1 is (A) p 1 (B) 2 (C) 1 (D) none of these 4. The number of ways of arranging six persons (having A, B, C and D among them) in a row so that A, B, C and D are always in order ABCD (not necessarily together) is (A) 4 (B) 10 (C) 30 (D) 720 5. Let A = {x : x is a prime number and x < 30}. The number of different rational numbers whose numerator and denominator belong to A is (A) 90 (B) 180 (C) 91 (D) none of these 6. Let S be the set of all functions from the set A to the set A. If n (A) = k, then n(s) is (A) k! (B) k k (C) 2 k 1 (D) 2 k 7. Let A be the set of 4-digit number a 1 a 2 a 3 a 4 where a 1 > a 2 > a 3 > a 4, then n(a) is equal to (A) 126 (B) 84 (C) 210 (D) none of these 8. There are three coplanar parallel lines. If any p points are taken on each of the lines, the maximum number of triangles with vertices at these points is (A) 3p 2 (p 1) + 1 (B) 3p 2 (p 1) (C) p 2 (4p 3) (D) none of these 9. In the next word cup of cricket there will be 12 teams, divided equally in two groups. Teams of each group will play a match againast each other. From each group 3 top teams will qualify for the next round. In this round each team will play against others once. Four top teams of this round will qualify for the semifinal round, where each team will play against the other three. Two top teams of this round will go to the final round, where they will play the best of three matches. The minimum number of matches in the next world cup will be (A) 54 (B) 53 (C) 38 (D) none of these 10. In a plane there are two families of lines y = x + r, y = x + r, where r {0, 1, 2, 3, 4}. The number of squares of diagonals of length 2 formed by the lines is (A) 9 (B) 16 (C) 25 (D) none of these 11. The number of flags with three strips in order, that can be formed using 2 identical red, 2 identical blue and 2 identical white strips is (A) 24 (B) 20 (C) 90 (D) 8 Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 59

12. The number of rational numbers p/q, where p, q {1, 2, 3, 4, 5, 6} is (A) 23 (B) 32 (C) 36 (D) none of these 13. A is a set containing n elements. A subset P of A is chosen the set A is reconstructed by replacing the elements of P. A subset Q of A is again chosen. The number of ways of choosing P and Q so that P Q contains exactly one elements is (A) 4 n (B) 3 n 1 (C) 3 n (D) n3 n 1 14. Triplet (x, y, z) is chosen from the set {1, 2, 3,...n}, such that x < y < z. The number of such triplets is (A) n 3 (B) n C 3 (C) n C 2 (D) n C 2 + n C 3 15. There are k different books and l copies of each in a college library. The number of ways in which a student can make a selection of one or more books is (A) (k + 1) l (B) (l + 1) k (C) (k + 1) l 1 (D) (l + 1) k 1 16. Along a railway line there are 20 stations. The number of different tickets required in order so that it may be possible to travel from every station to every station is (A) 380 (B) 225 (C) 196 (D) 105 17. The number of ways in which 4 letters of the word MATHEMATICS can be arranged is given by (A) 136 (B) 192 (C) 1680 (D) 2454 18. Let y be an element of the set A = {1, 2, 3, 5, 6, 10, 15, 30} and x 1, x 2, x 3 be positive integers such that x 1 x 2 x 3 = y, then the number of positive integral solutions of x 1 x 2 x 3 = y is (A) 64 (B) 27 (C) 81 (D) none of these 19. The sum of the factors of 7!, which are odd and are of the form 3t + 1 where t is a whole number, is (A) 10 (B) 8 (C) 9 (D) 15 20. Let S be the set of 6-digits a 1 a 2 a 3 a 4 a 5 a 6 a 7 (all digits distinct) where a 1 > a 2 > a 3 > a 4 < a 5 < a 6. Then n(s) is equal to (A) 210 (B) 2100 (C) 4200 (D) 420 21. The number of positive integral solutions of the equation x 1 x 2 x 3 =60 is (A) 54 (B) 27 (C) 81 (D) none of these 22. Consider a set {1, 2, 3,..., 100}. The number of ways in which a number can be selected from the set so that it is of the form x y, where x, y, N and 2, is (A) 12 (B) 16 (C) 5 (D) 11 23. The number of n-digit numbers, no two consecutive digits being the same, is (A) n! (B) 9! (C) 9 n (D) n 9 Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 60

24. The number of divisors of 3630, which have a remainder of 1 when divided by 4, is (A) 12 (B) 6 (C) 4 (D) none of these 25. The number of ways of selecting two numbers from the set {1, 2,..., 12} whose sum is divisible by 3 is (A) 66 (B) 16 (C) 6 (D) 22 26. Number of positive integral solutions of x 1 x 2 x 3 = 30 is (A) 27 (B) 6 (C) 9 (D) 18 Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 61

SUBJECTIVE TYPE 1. Two n-digit integers (leading 0 allowed) are said to be equivalent if one is a permutation of the other. Thus 10075 and 01057 are equivalent. Find the number of 5-digit integers such that no two are equivalent. 2. Use a combinatorial argument to prove that (2n 1)! ( n C 1 ) 2 + 2 ( n C 2 ) 2 + 3 ( n C 3 ) 2 +... + n ( n C n ) 2 = (( n 1)! ) 2 3. Let T(n) denote the number of non-congruent triangles with integer side lengths and perimeter n. Thus T(1) = T(2)=T(3)=T(4) = 0, while T(5) = 1. Prove that:- i) T(2006) < T(2009) ii) T(2005) = T(2008) 4. Find the number of different ways of painting a cube by using a different colour for each face from six available colours. (Any two colour schemes are called different if one cannot coincide with the other by a rotation of the cube.) 5. Take a ABC. Take n points of sub-division on side AB and join each of them to C. Likewise, take n points of sub-division on side AC and join each of them to B. Into how many parts is ABC divided? 6. A positive integer n has the decimal representation n = d 1 d 2... d m. (a) n is called ascending if 0 < d 1 d 2... d m (b) n is called strictly ascending if 0 < d 1 < d 2 <... < d m. Find the total number of type (a) and type (b) numbers, which are less than 10 9. 7. Prove that n r 1 n P = [n! e 1], where [ ] is the step function. r (Hint : You will need the series representation of e.) 8.. Show that the combinatorial number 2n C n is even for all n. 9. Show that 2 3n. 3 n divides (4n!) for all n. 10. Prove that 2 n divides (n + 1) (n + 2)... (2n) for all n. 11. Prove (combinatorially) that k n 1 0 EXERCISE 3 (B) 2 k 2 n 1. (Hint : Let 1 m n. How many subsets of {1, 2,..., n} have m as the maximal element?) 12. Let N(k) = {1, 2,..., k}. Find the number of :- i) functions from N(n) to N(m) ; ii) one-to-one functions from N(n) to N(m), n m. iii) strictly increasing functions from N(n) to N(m), n m iv) non- decreasing functions from N(n) to N(m). 13. Let 1 n r. The Stirling number of the first kind, S(r, n), is defined as the number of arrangements of r distinct objects around n identical circular tables so that each table contains atleast one object. a) Show that :- i) S(r, 1) = (r 1)! ; ii) S(r, r 1) = r C 2, r 2 ; Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 62

WINDOW TO I.I.T. - JEE Q.1 How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions? [JEE-2000] (A) 16 (B) 36 (C) 60 (D) 180 Q.2 Let T n denote the number of triangles which can be formed using the vertices of a regular polygon of ' n ' sides. If T n + 1 T n = 21, then ' n ' equals: [JEE-2001] (A) 5 (B) 7 (C) 6 (D) 4 Q.3 The number of arrangements of the letters of the word BANANA in which the two N s do not appear adjacently is [JEE-2002] (A) 40 (B) 60 (C) 80 (D) 100 Q.4 Number of points with integral co-ordinates that lie inside a triangle whose co-ordinates are (0, 0), (0, 21) and (21,0) [JEE-2003] (A) 210 (B) 190 (C) 220 (D) None (n )! Q.5 Using permutation or otherwise, prove that n (n!) 2 is an integer, where n is a positive integer. [JEE-2004] Q.6 A rectangle with sides 2m 1 and 2n 1 is divided into squares of unit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is (A) (m + n + 1) 2 (B) 4 m + n 1 (C) m 2 n 2 (D) mn(m + 1)(n + 1) [JEE-2000] Q.7 If r, s, t are prime numbers and p, q are the positive integers such that their LCM of p, q is is r 2 t 4 s 2, then the numbers of ordered pair of (p, q) is (A) 252 (B) 254 (C) 225 (D) 224 [JEE-2006] Q.8 The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is [JEE-2007] (A) 360 (B) 192 (C) 96 (D) 48 Q.9 Consider all possible permutations of the letters of the word ENDEANOEL Match the statements / Expression in Column-I with the statements / Expressions in Column-II. [JEE-2008] Column-I Column-II (A) The number of permutations containing the word ENDEA is (P) 5! (B) The number of permutations in which the letter E occurs in the (Q) 2 5! first and the last position is (C) The number of permutations in which none of the letters D, L, N (R) 7 5! occurs in the last five positions is (D ) The number of permutations in which the letters A, E, O occurs (S) 21 5! only in odd positions is Q.10 The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is [JEE-2009] (A) 55 (B) 66 (C) 77 (D) 88 Head Office : Andheri : 26245223 : MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW # 63