Permutations and Combinations: IITJEE Mains problem solving practice

Problem 1:

Find all natural numbers n such that $n!$ ends with exactly 26 zeros.

Problem 2:

Find the largest two digit prime that divides $200 \choose 100$ .

Problem 3:

Find all natural numbers $n \leq 14$ such that $n! + (n+1)!+(n+2)!$ is divisible by 25.

Problem 4:

When $30!$ is computed, it ends in 7 zeros. Find the digit that immediately precedes these zeros.

Problem 5:

Prove that $\frac{(n^{2})!}{((n)!)^{n+1}}$ is a natural number for all $n \in N$ .

Cheers,

Nalin Pihwa.

Mathematics Hothouse