What's The Probability?

Probability Level 3

For fixed integers nn and kk, find all kk-tuples of non-negative integers (a1,a2,ak) (a_1, a_2, \ldots a_k) such that

a1+a2++ak=n a_1 + a_2 + \cdots + a_k = n

For each integer i i from 0 to nn, let pi p_i be the probability that one of the aj a_j is equal to ii . What is the value of

1×p1+2×p2+n×pn? 1\times p_1 + 2 \times p_2 + \cdots n \times p_n ?

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