Find the number of ways in which 7 identical books can be distributed among 4…

Concept: When n identical items are distributed among r distinct groups, and a group may receive zero, one, or more items, the number of ways is the combinations-with-repetition count: (n + r − 1) choose (r − 1).

Application:

1. Here the 7 books are treated as identical items, so n = 7, and the 4 people A, B, C, D are the r = 4 distinct groups.

2. Substituting into the formula gives (7 + 4 − 1) choose (4 − 1), i.e. 10C3.

3. 10C3 = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120.

Cross-check: This matches the equivalent stars-and-bars picture — arranging 7 identical stars (books) and 3 dividers (bars) among them, in a row of 10 symbols, gives 10! / (7! × 3!) = 120 distinct arrangements, the same count arrived at above.

So there are 120 ways to distribute the 7 books among the 4 people.