The notation $n!$ (read "n factorial") is defined as the product of the first n positive integers. (For example, $3! = 1 \cdot 2 \cdot 3 = 6$.) Define the superfactorial of a positive integer, denoted by $n!^!$, to be the product of the factorials of the first n integers. (For example, $3!^! = 1! \cdot 2! \cdot 3! = 12$.) How many factors of 7 appear in the prime factorization of $50!^!$, the superfactorial of 50 ?

A. 147

B. 150

C. 156

D. 163

E. 171

1. 2026 AMC 8 官方解析

$50!^! = 1! \times 2! imes 3! imes \cdots imes 49! imes 50!$

$1! \sim 6!$ ，无7

$7! \sim 13!$ ，各1个7，$7 \times 1$个

$14! \sim 20!$ ，各2个7，$7 \times 2$个

$21! \sim 27!$ ，各3个7，$7 \times 3$个

$28! \sim 34!$ ，各4个7，$7 \times 4$个

$35! \sim 41!$ ，各5个7，$7 \times 5$个

$42! \sim 48!$ ，各6个7，$7 \times 6$个

$49! \sim 50!$ ，各8个7，$2 \times 8$个

共：$7 \times (1 + 2 + 3 + 4 + 5 + 6) + 2 imes 8 = 147 + 16 = 163$ （个）