# Solution: 2011-3 Counting functions

Let us write $$[n]=\{1,2,\ldots,n\}$$. Let $$a_n$$ be the number of all functions $$f:[n]\to [n]$$ such that $$f([n])=[k]$$ for some positive integer $$k$$. Prove that $a_n=\sum_{k=0}^{\infty} \frac{k^n}{2^{k+1}}.$

The best solution was submitted by Kang, Dongyub (강동엽), 전산학과 2009학번. Congratulations!

Here is his Solution of Problem 2011-3.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 박민재 (2011학번, +3), 김치헌 (수리과학과 2006학번, +2), 이동민 (수리과학과 2009학번, +2), 구도완 (해운대고등학교 3학년, +2).

P.S. A common mistake is to assume that $$\sum_{i}\sum_{j}$$ can be swapped without showing that a sequence converges absolutely.

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