Permutations

From the given conditions it is obvious that a6=1. Now observe that for every selection of 5 numbers from {2,3,4...12}, there is only one way of arranging them as a1,a2,a3...a5 and corresponding to that selection, there is only one arrangement ofa7,a8.... Hence all we have to find is the number of ways of selecting 5 things from 11 things.

These inequalities indicate, that a6 is the smallest number (as all the other numbers are bigger than that). Therefore, the value of a6 is fixed (always =1).

This means, that we have to find the number of the relevant permutations of the rest of the numbers (in this case, 11 numbers left).

If we choose any five of them for {a1, a2, a3, a4, a5} and put them into a descending order, we get the only relevant permutation for this choice of five numbers. The same applies for the remaining six numbers, put into an ascending order to get the only one relevant permutation for the values of {a7, a8, a9, a10, a11, a12}.

Therefore, the solution is the number of the ways we can choose 5 values out of 11, that is nCr(11,5) = 462.