Question

令 $(a_1, a_2, a_3, \ldots, a_{12})$ 是 $(1, 2, 3, \ldots, 12)$ 的一个排列，满足且这样的排列的一个例子是 $(6, 5, 4, 3, 2, 1, 7, 8, 9, 10, 11, 12)$ 。求这类排列的个数。 $a_1 \gt a_2 \gt a_3 \gt a_4 \gt a_5 \gt a_6$ $\begin{aligned} &a_6 \lt a_7 \lt a_8 \lt a_9 \\ &\lt a_{10} \lt a_{11} \lt a_{12}. \end{aligned}$

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.

答案

答案：462

知识点：有限制的排列组合

难度评级：2180

解答：

项 $a_6$ 小于这个排列中的其他所有项，所以 $a_6 = 1$ 。现在选择剩余 $11$ 个数中哪些占据位置 $1$ 到 $5$ ：它们必须按递减顺序出现，所以排列方式被确定；其余六个数必须按递增顺序填入位置 $7$ 到 $12$ ，这也被确定。

每一种五个数的选择都给出唯一一个合法排列，所以总数为 $\binom{11}{5} = 462$ 。

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 第 4 题

答案：462

难度评级：2180

解答：

其他年份的第 4 题