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# Math OERWeek 8 Homework, Part B

The greatest reward for a student is not a good grade. It is the willingness of his teacher to listen to him.

- Nikolay Konstantinov

Answer every question. Try being nice to your eyes and posture by printing this page and working with pencil and paper. Then use the button at the bottom of the page to create a code by processing your answers. Copy-and-paste the code into an e-mail along with the answer to your short answer question.

Keep trying each homework assignment until you get 8 out of 10 or more.

For this homework it will help to refer to our chart about the four types of arrangements.

 n Ordered Items n Unordered Items remember that arrangements have no repetition allowed Use Only r Items Use All Items Use Only k Items Use All Items An Arrangement is Called Partial Permutation Complete Permutation Combination Trivial Way to Write How Many Possible Arrangements P(n,r)n permute r n!n factorial C(n,k)n choose k 1 Real-Life Example who earns 1st, 2nd, 3rd place in a contest put books on a library shelf pick a few people from a group everyone gets their hand stamped Formula n! ÷ (n−r)! n! n! ÷ (n−k)! ÷ k! 1

1. A board game store wants to display its six best-selling games in a window display. In how many different orders can it do this?

6 permute 1 = 6 6 factorial = 1 6 choose 6 = 1
6 permute 6 = 1 6 factorial = 30 6 choose 1 = 6
6 permute 6 = 30 6 factorial = 720 6 choose 1 = 30

2. At that board game store, eight customers arrive for a game night. They will be seated at two tables with four chairs, to play two different board games. In how many ways can these eight people be arranged as two groups of four? (Hint: find how many ways four of the eight people can be picked for the first table. Then you are done, because the rest will automatically be the people at the other table.)

8 permute 2 = 56 8 factorial = 40,320 8 choose 2 = 28
8 permute 4 = 1,680 (8 − 4)! = 24 8 choose 4 = 70

3. The four players at the first table will play a game where one player at a time is eliminated until only one victor remains. They decide that whomever is eliminated first buys snacks for their table, and whomever is eliminated second buys sodas. In how many ways can this happen?

4 permute 2 = 12 4 factorial = 24 4 choose 2 = 6
4 permute 4 = 24 (4 − 2)! = 2 4 choose 4 = 1

4. In a workshop with 11 participants, two participants are randomly chosen to write on the whiteboard. In how many ways can this happen?

22 55 110 362,880 19,958,400

5. Five kids are playing a keep-away game that starts with two being randomly chosen to stand at the far edges of the playground. (These two pass a ball to each other. The other three kids stand in the middle and try to intercept the ball.) In how many ways can two kids be randomly picked?

6 10 20 60 120

6. Continuing the previous problem, two of the five kids are sisters. What is the probability that those two sisters are the two kids randomly chosen?

1% 2% 5% 10% 17%

7. At recess five kids played with a soccer ball and plastic cones. They forget to bring their equipment back to the equipment closet. The recess monitor randomly picks one of them to get the ball, and a second to fetch the cones. In how many ways can this happen?

6 10 20 60 120

8. Continuing the previous problem, two of the five kids are sisters. What is the probability that those two sisters are the two kids randomly chosen?

1% 2% 5% 10% 17%

9. 15 permute 3 =

15 × 12 15 × 3! 15 × 14 × 13
15 × 3 15! ÷ 3! 15 × 14 × 13 × 12

10. 15 choose 3 =

the previous answer divided by 3
the previous answer divided by 6
the previous answer divided by 12
the previous answer divided by 15

Short Answer Questions:

A. Explain the difference between combinations and permutations. Why are probabilities for card games usually calculated with combinations?

B. For any given values of n and r = larger, P(n,r) or C(n,r)? Explain why. Can they ever be equal?

C. Why are probabilities always between 0 and 1? Why can't a probability be negative, or 12?