Let X and Y denote nonempty finite sets such that
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Let X and Y denote nonempty finite sets such that

Let X and Y denote nonempty finite sets such that |X| ≥ │Y│.

(i) Using inclusion/exclusion, find a formula for the number of surjective functions from X to Y. Express your answer in terms of n = |X| and k = |Y|.

(ii) Express your answer to (i) in terms of Stirling numbers.

Hint
Mathematicsii)The coefficients of expansions of falling and rising factorials, often known as the Pochhammer symbol, are expressed as polynomials using Stirling numbers. Having coefficients for the rising factorial that are (signed) Stirling numbers of the first kind....
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