More with Combinations

Transcription

1 Algebra II Wilsen BLOCK 5 Unit 11: Probability Day Two More with Combinations Example 1 A standard deck of 52 playing cards has 4 suits with 13 different cards in each suit. How many 5-card hands (assuming that the order in which they re dealt is unimportant) exist for each of the following? (a) any 5 cards (b) all 5 the same color (c) exactly 3 aces (d) at least 3 aces (e) no more than four red cards (f) exactly 2 are of the same suit (g) 2 are of the same suit, and 3 are of another suit

2 Example 2 a) How many 5 card poker hands could be formed? b) A royal flush is a straight flush with an ace as the high card. How many can be formed? c) How many straight flushes (make sure not to include the royal flushes)? d) How many four of a kind? e) How many straights?

3 1) When choosing an organic farm to work on for the summer, it is important to research and then narrow down your choices before contacting the farms for more information. Hayley wants to narrow the choices down from 12 farms to 7 farms. How many ways can a group of 7 farms be selected from the 12? 2) A basket from one of the farms contains 4 acorn squash, 5 gourds, and 8 pumpkins. How many ways can 2 squash, 1 gourd, and 2 pumpkins be chosen? 3) Peter decides he is going to make his fortune selling used microwaves. In his latest shipment of 17 microwaves, 15 work and 2 are broken. If he puts them all out on display anyway and a client comes in to buy 3 microwaves for her office, in how many ways can she choose: a) 3 microwaves? b) All good microwaves? c) Exactly two bad microwaves? d) At least one bad microwave? 4) Write all permutations of two of the letters D, F, and M, and then write all the combinations of two of the letters D, F and M. 5) A sandwich shop offers 7 different kinds of breads, 4 different kinds of spreads, 5 different kinds of meat, and 13 different kinds of cheese. If you order a sandwich and select one of each option, how many different sandwich possibilities are there?

4 6) Harry s divination class has 8 girls and 6 boys. a) The boys want to line up. How many different linear permutations are there? b) The class wants to line up, but Ron, Harry and Hermione must stand next to each other according to their height (tallest to shortest). How many different linear permutations are there? c) How many different linear permutations of the class are there in which a girl would be in the first 3 positions in line? d) Ms. McGonagall wants to create a textbook committee from the members of the students in the class mentioned above. If the committee must have six members, find the number of possible committees that would consist of: i. 2 boys and 4 girls ii. all boys iii. exactly 4 boys iv. AT LEAST 4 boys 7) How many different arrangements are there of the letters in the word: a. LOGARITHM? b. HARASSES? c. In how many ways can you arrange the letters of the word HARASSES, so that it begins with the word SEA?

5 8) Solve algebraically: (a) P(n, 5) = 576! C(n" 1, 3) (b) n P 4 = 20 n!1 C 2 (c) Solve for n. n P 4 = 8( n P 3 ) 9) Write all the combinations two of the letters D, F, and M. 10) From a group of 4 men and 8 women, how many committees of 2 men and 2 women can be formed? 11) In how many 5-card hands are all 5 cards of the same color?

6 12) How many four-digit numbers can be formed under the following conditions? a) The leading digit cannot be zero. b) The leading digit cannot be zero and no repetition of digits is allowed. c) The leading digit cannot be zero and the number must be a multiple of 5. (Hint: list a few numbers that are multiples of 5 to see what they have in common.) d) The number is at least ) Ms. Wilsen has an selection of 12 different spices in her pantry at home. She wants to finally organize them! In how many ways can she do the following? a) arrange the 12 different spices on a single rack? b) arrange the spices on a rack in such a way that 3 of the spices oregano, basil, and parsley are kept together?

7 14) Find the number of 4-card hands (from a 52-card deck) that contain the cards specified. a) 4 face cards (kings, queens, or jacks) b) 2 kings and 2 other cards that are not kings c) 4 spades or 4 clubs d) At most 1 king e) At least 1 heart f) 4 red cards or 4 face cards 15) The window of a music store has 8 stands in fixed positions where instruments can be displayed. In how many ways can 3 identical guitars, 2 identical keyboards, and 3 identical violins be displayed?

8 16) Ms. Wilsen has gotten tired of grading, and just decides to give out grades randomly this semester, and places the integers 60 to 100 in a hat. (a) If one student picks out his or her grade, records it, and puts it back before the next student takes a grade, in how many ways can 5 students draw an odd numbered grade? (b) If a student picks out a grade and keeps it before the next one takes a grade, in how many ways can 5 students draw an odd number? 17) A box of chocolates contains 9 dark chocolates, 6 milk chocolates, and 4 white chocolates. How many ways can 5 chocolates be selected to meet each condition? (a) All are milk chocolates. (b) All are white chocolates. (c) 2 are white chocolates, 2 are milk chocolates, 1 is dark chocolate. (d) At least 3 are dark chocolate?

9 Answers: 1) 792 2) 840 3a) 680 b) 455 c) 15 d) 225 5) a) 720 b) 479,001,600 c) 1.34 x 10^10 di) 1050 dii) 1 diii) 420 div) 469 7a) 362,880 b) 3360 c) 60 8a) 12 b) 5 c) 11 10) ) 131,560 12a) 9000 b) 4536 c) 1800 d) a) 479,001,600 b) 21,772,800 14a) 495 b) 6768 c) 1430 d) 263,764 e) 188,474 f) 15,430 15) a) 3,200,000 b) 1,860,480 17a) 6 b) NA c) 810 d) 5166