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# PS02 - MATH 681 Problem Set#2 Show work for each problem...

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MATH 681 Problem Set #2 Show work for each problem. Answers without justification, or justified solely by direct enumeration, are not acceptable. Arithmetic expressions may be left unsimplified. This problem set is due at the beginning of class on September 17 . 1. (10 points) Prove the following identity combinatorially: n X i =0 i n i = n 2 n - 1 2. (10 points) If a fair coin is flipped n times, what is the probability that (a) (3 points) The first head comes after exactly m tails? (b) (7 points) The i th head comes after a total of m previous tails? 3. (10 points) Using paths through a lattice (or some other combinatorial object, if you prefer), prove that the following identity is true for any m k n : m + n n = m X i =0 k i m + n - k m - i 4. (10 points) Construct generating functions for the number of nonnegative integer solutions to the following equations:
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