Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
The number of ways of distributing 10 different books among 4 students ($S_1, S_2, S_3$ and $S_4$) such that $S_1$ and $S_2$ get 2 books each and $S_3$ and $S_4$ gets 3 books each is:
12600
25200
$^{10}C_4$
$\frac{10!}{2!2!3!3!}$
Step-by-Step Solution
Key Concept: The multinomial coefficient $\frac{n!}{n_1!n_2!...n_k!}$ directly counts ordered partitions of distinct objects into labeled groups.
We need to distribute 10 different books among 4 students with specific constraints: $S_1$ gets 2, $S_2$ gets 2, $S_3$ gets 3, $S_4$ gets 3 books. Since the books are different and students are distinct, we use the multinomial coefficient. First, choose 2 books for $S_1$: $\binom{10}{2}$; then 2 for $S_2$ from remaining 8: $\binom{8}{2}$; then 3 for $S_3$ from remaining 6: $\binom{6}{3}$; finally 3 for $S_4$: $\binom{3}{3}$. This equals $\frac{10!}{2!\cdot 2!\cdot 3!\cdot 3!} = \frac{3628800}{2\cdot 2\cdot 6\cdot 6} = \frac{3628800}{144} = 25200$.
Correct Answer: 2,4