Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
The number of ways in which 8 distinguishable apples can be distributed among 3 boys such that every boy should get atleast 1 apple and atmost 4 apples is $K \cdot P_r$ where K has the value equal to:
Step-by-Step Solution
Key Concept: Combinatorial identities like the hockey-stick identity and Pascal's identity enable collapsing large sums of binomial coefficients into simple closed forms.
We expand the given sum involving binomial coefficients with alternating indices. By grouping and reorganizing the terms using the hockey-stick identity and properties of binomial coefficients, we obtain $^{n+1}C_{m+1} + 2(^nC_{m+2} + ^{n-1}C_{m+1} + \ldots) = ^{n+1}C_{m+1} + 2\cdot^{n+1}C_{m+2} = ^{n+1}C_{m+2} + ^{n+2}C_{m+2}$.
Correct Answer: 4