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Unformatted text preview: 1 ISyE 3770 – Fall 2007 Homework #2 (Covers Modules 6–9) — Solutions 1. Suppose that 20 clowns are trying to get into a small car that will only accommodate a maximum of 12 clowns. How many possible choices of 12 clowns are there? Solution: ˆ 20 12 ! 2 2. How many ways can you arrange the letters in “TENNESSEE”? Solution: This is a multinomial combination. ˆ 9 1 , 4 , 2 , 2 ! = 9! 1!4!2!2! = 3780 . 2 3. A bridge hand contains 13 cards from a standard deck. Find the probability that a bridge hand will contain (a) Exactly 2 aces. Solution: ˆ 4 2 !ˆ 48 11 ! ˆ 52 13 ! 2 (b) At least 2 aces. Solution: 4 X i =2 ˆ 4 i !ˆ 48 13 i ! / ˆ 52 13 ! 2 (c) 8 spades. Solution: ˆ 13 8 !ˆ 39 5 ! ˆ 52 13 ! 2 (d) 8 cards of the same suit. Solution: 4 ˆ 13 8 !ˆ 39 5 ! ˆ 52 13 ! 2 4. A die is thrown 7 times. Find (a) Pr (‘6’ comes up at least once). Solution: 1 Pr (no 6’s appear) = 1 (5 / 6) 7 2 . 2 (b) Pr (each face appears at least once). Solution: Denote the six faces by A,B,C,D,E,F. Thus, we need to find the number of tosses of the form A,A,B,C,D,E,F. We then see that i. The # ways to choose A is 6. ii. The # ways to place A is ˆ 7 2 ! . iii. The # ways to permute B,C,D,E,F is 5!. iv. The # ways to toss the die 7 times is 6 7 . Thus, Pr (each face appears at least once) = 6 · ˆ 7 2 ! · 5! / 6 7 . 5. Write a computer program in your favorite language to calculate combinations. Demonstrate your program on C 100 , 50 ....
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 Fall '07
 goldsman
 Conditional Probability, Probability theory, Joe misses

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