Find the number of ways in which 5 toffees can be distributed over 4 different…

Concept: Distributing n identical items among r distinct people (any person may get zero or more) is a stars-and-bars problem: the number of ways equals C(n + r - 1, r - 1) — arrange n stars and (r - 1) bar-dividers in a row and choose where the bars go.

Application

1. Identify n = 5 (toffees, identical) and r = 4 (people A, B, C, D, distinct).

2. Substitute into the formula: number of ways = C(n + r - 1, r - 1) = C(5 + 4 - 1, 4 - 1) = C(8, 3).

3. Expand C(8, 3) = 8! / (3! × 5!) = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = 56.

Cross-check: C(8, 3) = C(8, 5) by the symmetry of combinations, and 8! / (5! × 3!) also gives 56 — the two computations agree, confirming the count.

So there are 56 ways to distribute the 5 toffees among A, B, C, and D.