Counting Problems for Group 1 (Due by EOC Feb. 27)

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1 Counting Problems for Group 1 (Due by EOC Feb. 27) Not All Things Must Pass. 1. Forty-one students each took three exams: one in Algebra, one in Biology, and one in Chemistry. Here are the results 12 failed the Algebra exam 5 failed the Biology exam 8 failed the Chemistry exam 2 failed Algebra and Biology 6 failed Algebra and Chemistry 3 failed Biology and Chemistry 1 failed all three How many students passed exams in all three subjects? {Hint: Make a Venn diagram.} Facebook Shmacebook. 2. After graduation exercises, each senior gave a snapshot of himself or herself to every other senior and received a snapshot in return. If 2,000,810 snapshots were exchanged, how many seniors were in the graduation class? Hint: You can use a combinations formula or start with small classes and look for a pattern. It All Adds Up To Something. 3. a) There are 120 five-digit numbers that use all the digits 1 through 5 exactly once. What is the sum of the 120 numbers? Hint: numbers How many of each digit occur in each column? b) If the digits can be repeated, then there are 3,125 five-digit numbers that can be formed. What is the sum of the 3,125 numbers? c) Repeat part a) with the digits 1 through 6. d) Repeat part b) with the digits 1 through 6.

2 A Checkered Past, Present, And Future. 4. a)how many squares can you find on a 8 8 checkerboard? b) an n n checkerboard? Hint: Start with smaller boards and look for a pattern. 1 large and 4 small = 5 1 large, 4 medium, and 9 small = 14 1 extra large, 4 large, 9 medium, and 16 small = 30

3 Man Or Woman, We Mean Business. 5. Out of 35 students in a math class, 22 are male, 19 are business majors, 27 are first-year students, 14 are male business students, 17 are male first-year students, 15 are first-year students who are business majors, and 11 are male first-year business majors. a) How many upper class female non-business majors are in the class? b) How many female business majors are in the class? Male Female Business First-year student Non-business Ups And Downs With And Without Nine Lives. 6. a) An elevator starts at the basement with 8 people(not including the elevator operator) and discharges them all by the time it reaches the 6 th floor. In how many ways could the operator record the number of people leaving the elevator on each of the 6 floors? b) If the same elevator also has 10 cats, in how many ways could the operator record the number of cats leaving the elevator on each of the 6 floors? c) In how many ways could the operator record the number of people and the number of cats leaving the elevator on each of the 6 floors? {Hint: See the hint for #4 for group 2.}

4 , so we get. This rearranges into na B na nb nu n A B, and since n A B 0, it must be that. This means that the number of elements in the intersection of Consider the sets A and B inside a universal set U. nu n A B n A B that nu na nb na B n A B n A B n A nb nu A and B is at least n A nb nu, and it also means that if n A nb nu 0, then it s possible that n A B 0. This result can be extended to the case of three sets as follows: n A B C n A B C n A nb C nu na nb nc nu nu so n A B C n A nb nc 2 nu. It can further be extended to the case of four sets as follows: na B C D n A B C D na nb C D nu, so n A n B n C n D 2 n U n U n A B C D n A nb nc nd 3 nu. In general, you can show that na1 A2 Ak na1 na2 nak k 1 nu. Also, n A B n A and n A B nb, so n A B min n A, nb you can show that n A A A min n A, n A,, n A. 1 2 k 1 2. In general, k Yes, We Have No Banana Sandwiches Today. 7. There are 24 children on a school outing. At lunchtime, 11 of them ate a sandwich, 9 of them ate a banana, and n of them ate neither a sandwich nor a banana. Find a) the smallest possible value of n. {Hint: See the previous discussion.} b) the largest possible value of n.

5 Hopefully, You ll Have A Lot Of Interest In These Banks. 8. Determine the number of different paths for spelling the word BANK: K N K A N K B A N K {Hint: The letters actually form a tree diagram: B A A N N N N } Multiples Of Multiples. 9. a) How many of the first 1,000 counting numbers are multiples of 2 or multiples of 5? {Hint: multiples of 2 or multiples of 5 multiples of 2 multiples of 5 nmultiples of 2 and 5 n n n.} b) How many of the first 10,003 counting numbers are multiples of 2 or multiples of 3? Man Have We Got A Lot Of Different Pizzas. 10. A pizza parlor offers four sizes of pizza(small, medium, large, and colossus), two types of crust(thick and thin), and 14 different toppings. a) If you must choose a size, a crust, and at least 1 topping, and you can t duplicate a topping, how many different pizzas can be made? b) How many if double toppings are allowed?

6 Trains, Buses, And Automobiles travelers were questioned about the method of transport they used on a particular day. Each of them used one or more of the methods shown in the Venn diagram. Of those questioned, 6 traveled by bus and train only, 2 by train and car only, and 7 by bus, train, and car. The number x who traveled by bus only was equal to the number who traveled by bus and car only. 35 people used buses, and 25 people used trains. Find: Bus x 6 Train x 7 2 Car U a) the value of x. b) the number who traveled by train only. c) the number who traveled by at least two methods of transport. d) the number who traveled by car only. Congratulations Mr. and Mrs. Zeta. 12. Mr. and Mrs. Zeta want to name their baby so that its monogram(first, middle, and last initials) will be in alphabetical order with no letters repeated. How many such monograms are possible?

7 Corporation Games. 13. A corporation employs 95 people in the areas of sales, research, and administration. 10 people can function in any of the three areas, 30 can function in sales and administration, 20 can function in sales and research, and 15 can function in administration and research. There are twice as many people in sales as in research, and the same number in sales as in administration. What are the possible numbers of people who can function in sales only, administration only, and research only? {Hint: Sales x Research 5 y z Administration w You get the equations: x 40 2 y 25 From these you can conclude that between 10 and 20., x 40 z 35, and x y z w y x 5, zx 5, w50 2 x, so x is even and 2 x 1 y x 5 2 zx 5 w x } Oddly, It s As Easy As 1, 2, 3, 4, a) From the digits 1, 2, 3, 4, 5, how many four-digit numbers with distinct digits can be constructed? b) Of these, how many are odd?

8 The Truth About Cats And Dogs And Birds And Fish. 15. A survey of 136 pet owners resulted in the following information: 49 own fish; 55 own a bird; 50 own a cat; 68 own a dog; 2 own all four; 11 own only fish; 14 own only a bird; 10 own fish and a bird; 21 own fish and a cat; 26 own a bird and a dog; 27 own a cat and a dog; 3 own fish, a bird, a cat, but no dog; 1 owns fish, a bird, a dog, but no cat; 9 own fish, a cat, a dog, but no bird; and 10 own a bird, a cat, a dog, but no fish. Bird Cat Fish Dog U a) How many of the surveyed pet owners have no fish, no birds, no cats, and no dogs? b) How many of the surveyed pet owners have exactly two of the pet types? c) How many of the surveyed pet owners have at least two of the pet types? Is It Broken If It s Out Of Alphabetical Order? 16. Find the number of 5 letter permutations of the letters A, B, C, D,, Z that are in alphabetical order. Appliances, Get Your Free Appliances! 17. A woman has 2 identical toasters and 3 identical blenders. Each day for five consecutive days, she gives away one of the 5 appliances. How many different ways can she do this? {Hint: Use the formula for permutations with duplicates, and use the position of the appliance as the day that it s given away.}

9 Imelda Returns. 18. A closet contains 10 pairs of shoes. If 8 shoes are selected, without replacement, how many ways can there be a) no complete pair? Hint: Here are the ten pairs of shoes: L10 R 10 10C Which 8 pairs? Shoe from 1 st pair selected Shoe from 2 nd pair selected Shoe from 8 th pair selected b) exactly one complete pair? Vote As Many Times As You Can. 19. In an election, each voter can distribute up to 5 votes among 6 candidates. For example, you could cast 3 votes for one candidate and 2 for another, or you could cast 1 vote for each of 4 candidates and not cast your fifth vote. In how many ways can you distribute your votes? {Hint: Candidate 1 Candidate 2 Candidate 3 Candidate 4 Candidate 5 Candidate 6 No one See the hint for #4 for group 2.}

10 More Than One Way To Get From A To B. 20. a) How many paths are possible from A to B if all movements must be to the right or down? A {Hint: See the hint for #17 for group 2.} b) How many paths are possible from A to B if all movements must be to the right or down, and you must pass through point C? A B C B