How many ways can 20 identical toys be divided among 10 childrens if each…

Answer: C(19, 9) = 92,378

Reasoning:

• Let xi be the number of toys given to child i. We need the number of integer solutions to x1 + x2 + ... + x10 = 20 with each xi ≥ 1.

• Give each child one toy first. Define yi = xi − 1, so each yi ≥ 0 and y1 + y2 + ... + y10 = 20 − 10 = 10.

• The number of nonnegative integer solutions to y1 + ... + y10 = 10 is C(10 + 10 − 1, 10 − 1) = C(19, 9).

• Therefore the number of ways to distribute the 20 identical toys so each child gets at least one is C(19, 9) = 92,378.