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I was studying linearity of expectation topic and there was an example of assigning n processes to n different servers randomly, as per i can think - 1st process to any of n servers so "n" ways, 2nd process to any of the n servers, so again "n" ways, in the same way nth process to any of the n servers, again in "n" ways. A total of n times n so n^2 ways(so the sample space), but the actual sample space said was n^n. How can it be? Thanks.

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Think of it as a tree. The first choice generates n branches, the second choice generates n branches from each of the original n branches (n^2 total), the third choice generates n branches from each of the n^2 branches (n^3 total), and so on until there are n^n branches. However, this is the wrong site. This site is about Mathematica software, not mathematics.

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Denote the set of servers as $S = \{s_1,s_2,\dots,s_n\}$.

An outcome of this experiment is a length-$n$ tuple belonging to the set $S^n$ (the set of all possible outcomes, aka sample space). Its cardinality is $|S|^n=n^n$ by the rule of product.

To make this a little clearer, we can express an outcome in the form $(p_1,p_2,\dots,p_n)$ where $p_i$ is the server who has been assigned process $p_i$.

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