Balls numbered 1 through N are in an urn. Suppose that n,n ≤ N, of them are randomly selec...

Balls numbered 1 through N are in an urn. Suppose that n,n ≤ N, of them are randomly selected without replacement. Let Y denote the largest number selected.

(a) Find the probability mass function of Y.

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(b) Derive an expression for E[Y] and then use Fermat’s combinatorial identity (see Theoretical Exercise of Chapter 1) to simplify the expression.

Exercise

The following identity is known as Fermat’s combinatorial identity:

Give a combinatorial argument (no computations are needed) to establish this identity.