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1. A composition of a positive integer n is an ordered list of positive integers (called parts) that sum to n. The four compositions of 3 are 3; 2,1; 1,2 and 1,1.1. (a) By considering ways to insert plus signs and commas in a list of n ones, obtain a formula for the number of compositions of n. Hint. The four coupositions above correspond to 1+1+1; 1+1,1; 1,1+1 and 1,1,1, respec- tively. (b) Prove that the average number of parts in a composition of n is (n + 1)/2 Hint. Reverse the roles of + and and then look at the number of parts in the original and role-reversed compositions
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