# Subset permutations

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, 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 $$P(n, r)$$?

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.