Permutations & Combinations
Permutations & Combinations
star_batch_jee_advanced_2025
Grade None
Question:
Number of ways in which four different toys and five indistinguishable marbles can be distributed between Amir, Aniket and Anthony, if each child receives at least one toy and one marble, is:
Step-by-Step Solution
Key Concept: Sum of binomial coefficients from the expansion of $(1+1)^{m+1}$ yields $2^{m+1}$.
Using the binomial theorem expansion $(1+x)^{m+1} = 1 + ^{m+1}C_1x + ^{m+1}C_2x^2 + ... + ^{m+1}C_nx^n$, we substitute $x=1$ to get $(1+1)^{m+1} = 2^{m+1} = 1 + ^{m+1}C_1 + ^{m+1}C_2 + ... + ^{m+1}C_n$. By setting $S_r = ^{m+1}C_r$ and using the recurrence relation, the sum of binomial coefficients equals $2^{m+1} - (n+1)$.
Correct Answer: 4