Ass2solns - MATH 352 Assignment#2 Solutions 1 Use the...

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MATH 352 Assignment #2 Solutions 1. Use the Principle of Inclusion/Exclusion to solve each of the following problems: (a) Find the probability that a permutation of a 1 , a 1 , a 2 , a 2 , . . . , a n , a n will have no identical symbols consecutive. The first and last positions are not considered to be consecutive. Solution: Let A i be the event that the two a i are consecutive. The desired probability is the probability that none of the events occur, ie. 1 - P ( n k =1 A i ) = 1 - n k =1 ( - 1) k - 1 S k . For any i , to calculate P ( A i ), ‘glue’ together the two a i , then permute the object a i a i together with the other 2 n - 2 objects: (2 n - 1)! / (2!) n - 1 ways. Therefore P ( A i ) = (2 n - 1)! 2 n - 1 / (2 n )! 2 n = 2 · (2 n - 1)! (2 n )! . This gives S 1 = n · 2 · (2 n - 1)! (2 n )! . Using this same method, for any choice of k events, P ( A i 1 A i 2 . . . A i k ) = (2 n - k )! 2 n - k / (2 n )! 2 n = 2 k · (2 n - k )! (2 n )! , and S k = n k · 2 k · (2 n - k )! (2 n )! . The desired probability is 1 - S 1 + S 2 - · · · ± S n . (b) Suppose each of n sticks is broken into one long and one short part. The 2 n parts are then shuffled and rearranged into n pairs from which new sticks are formed. Find the probabilitiy that i. the parts will be joined into their original form. Solution: Label the short parts of the sticks s 1 , s 2 , . . . , s n , and their corresponding long parts l 1 , l 2 , . . . , l n .
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