JEE Main 2025 Apr 2 Shift 1, Mathematics Q1: Permutations And Combinations

The highest power of a prime $p$ dividing $N!$ is given by:

$$E_p(N!)=\sum _{k=1}^{\infty }\lfloor \frac{N}{p^k}\rfloor$$

For $p=3$ and $N=50$, only the terms with $3^k\le 50$ contribute:

$$E_3(50!)=\lfloor \frac{50}{3}\rfloor +\lfloor \frac{50}{9}\rfloor +\lfloor \frac{50}{27}\rfloor +\lfloor \frac{50}{81}\rfloor$$

$$=16+5+1+0=22$$

$\therefore$ The largest $n$ for which $3^n\mid 50!$ is $22$. Option (2) is correct.