Assignment3Solution.pdf - CSC 226 Summer 2016 Assignment 3...

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CSC 226: Summer 2016 Assignment 3 Solution July 18, 2016 Answer 1 f ( n, m, k, i ) = k + i k m - k - 1 + n - i m - k - 1 According to the figure, n = m = 10 , k = 4 f ( n, m, k, i ) = 4 + i 4 10 - 4 - 1 + 10 - i 10 - 4 - 1 = 4 + i 4 15 - i 5 Answer 2 The answer is 0. From the binomial theorem we know that, ( a + b ) n = n 0 a 0 b n + n 1 a 1 b n - 1 + n 2 a 2 b n - 2 + ... + n n a n b n - n = n X k =0 n k a k b n - k Let a = - 1 , b = 1. Then the equation becomes, (1 - 1) n = n X k =0 n k ( - 1) k (1) n - k = n X k =0 n k ( - 1) k = 0 1
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Answer 3 According to the hockey stick principle, n X i = r i r = n + 1 r + 1 Applying the principle, 12 5 + 11 5 + 10 5 + 9 5 + 8 5 = 12 X i =5 i 5 - 7 X i =5 i 5 = 13 6 - 8 6 Answer 4( a ) Let X = the event when we first have a TH Then, E [ X ] = the expected number of flips to X We first flip the coin, the probability of getting a H is 1 / 2 and the probability of getting a T is also 1 / 2 E [ X ] = Pr (
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