2026 AMC 8 Problem 24

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 51^!, the superfactorial of 51?

\textbf{(A)} 147 \qquad \textbf{(B)} 150 \qquad \textbf{(C)} 156 \qquad \textbf{(D)} 168 \qquad \textbf{(E)} 171

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.