2006 AIME II Problem 4

Question

Let

(a_1, a_2, a_3, \ldots, a_{12})

be a permutation of

(1, 2, 3, \ldots, 12)

for which

a_1 \gt a_2 \gt a_3 \gt a_4 \gt a_5 \gt a_6

and

\begin{aligned} &a_6 \lt a_7 \lt a_8 \lt a_9 \\ &\lt a_{10} \lt a_{11} \lt a_{12}. \end{aligned}

An example of such a permutation is

(6, 5, 4, 3, 2, 1, 7, 8, 9, 10, 11, 12).

Find the number of such permutations.

Answer

a_6

is less than all eleven other terms, so

a_6 = 1.

Once you choose which five of the remaining

11

numbers fill the first five slots, the order of both halves is forced.

The term

a_6

is smaller than every other term of the permutation, so

a_6 = 1.

Now choose which five of the remaining

11

numbers occupy positions

1

through

5:

they must appear in decreasing order, so their arrangement is forced, and the other six numbers must fill positions

7

through

12

in increasing order, which is also forced.

Every choice of the five numbers gives exactly one valid permutation, so the count is

\binom{11}{5} = 462.

Problem 4 in other years