Question

Let $(a_1, a_2, a_3, \ldots, a_{12})$ be a permutation of $(1, 2, 3, \ldots, 12)$ for which $a_1 \gt a_2 \gt a_3 \gt a_4 \gt a_5 \gt a_6 \quad \text{and} \quad a_6 \lt a_7 \lt a_8 \lt a_9 \lt a_{10} \lt a_{11} \lt a_{12}.$ 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.

答案

Solution:

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 $\binom{11}{5} = 462.$

2006 AIME II Problem 4

Solution:

Problem 4 in Other Years