# What's The Probability?

Probability Level 3

For fixed integers $n$ and $k$, find all $k$-tuples of non-negative integers $(a_1, a_2, \ldots a_k)$ such that

$a_1 + a_2 + \cdots + a_k = n$

For each integer $i$ from 0 to $n$, let $p_i$ be the probability that one of the $a_j$ is equal to $i$. What is the value of

$1\times p_1 + 2 \times p_2 + \cdots n \times p_n ?$

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