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Given that $B = (a_1, a_2 \cdots, a_{12})$ is a permutation of the set $(1, 2, \cdots, 12)$ such that $a_1>a_2>a_3>a_4>a_5>a_6$ and $a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}$, then how many possible permutations can B represent?


I know of the permutation formula $P(n, r) = \frac{n!}{(n-r)!}$, but I don't know how that can be used here. How can I use $P(n, r)$?

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Firstly, you can see that $a_6$ is the minimum since it is smaller than any other nubmer in $B$. So, $a_6=1$. Now, choose sets $\{a_1,a_2,a_3,a_4,a_5\}$ and $\{a_7,a_8,a_9,a_{10},a_{11},a_{12}\}$. This you can do in $\binom{11}{5}=\binom{11}{6}$ ways. Once you have defined these two sets there is a unique way to put them in the array $B$ and that is to sort them appropriately.

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