# 2015 UNCO Math Contest II Problems/Problem 10

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## Problem $\begin{tabular}[t]{|c|c|c|c|}\hline & & & \\\hline & & & \\\hline \end{tabular}$

(a) You want to arrange $8$ biologists of $8$ different heights in two rows for a photograph. Each row must have $4$ biologists. Height must increase from left to right in each row. Each person in back must be taller than the person directly in front of him. How many different arrangements are possible? $\begin{tabular}[t]{|c|c|c|c|c|c|}\hline & & & & & \\\hline & & & & & \\\hline \end{tabular}$

(b) You arrange $12$ biologists of $12$ different heights in two rows of $6$, with the same conditions on height as in part (a). How many different arrangements are possible? Remember to justify your answers.

(c) You arrange $2n$ biologists of $2n$ different heights in two rows of $n$, with the same conditions on height as in part (a). Give a formula in terms of $n$ for the number of possible arrangements.

## Solution

(a) $14$ (b) $132$


(c) $\frac{1}{n + 1} \binom{2n}{n}$ $= \binom{2n}{n}- \binom{2n}{n-1}= \binom{2n}{n}- \binom{2n}{n+1}=\frac{1}{2n+1} \binom{2n+1}{n}=\frac{(2n)!}{(n+1)!n!}$