For positive integer $n$, the factorial notation $n!$ represents the product of the integers from $n$ to 1. (For example, $6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$.) What value of $N$ satisfies the following equation?
$5! \cdot 9! = 12 \cdot N!$

A. 10

B. 11

C. 12

D. 13

E. 14

1. 2020 AMC 8 官方解析

The given equation implies that
$N! = \frac{5! \cdot 9!}{12} = \frac{(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1) \cdot (9!)}{12} = 10 \cdot 9! = 10!$
so $N = 10$.

由所给的方程可以推断出：
$N! = \frac{5! \cdot 9!}{12} = \frac{(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1) \cdot (9!)}{12} = 10 \cdot 9! = 10!$
故$N = 10$ .