In how many ways can we distribute 10 pencils to 4 children so each child gets…

Solution:

We need the number of ways to distribute 10 identical pencils to 4 distinct children so that each child gets at least one pencil.

Key idea: Use the stars-and-bars method to count positive integer solutions.

• Let x1, x2, x3, x4 be the number of pencils given to each child. We require x1 + x2 + x3 + x4 = 10 with each xi ≥ 1.

• The number of positive integer solutions of x1 + x2 + x3 + x4 = 10 is C(10 - 1, 4 - 1) = C(9, 3).

• Compute C(9,3) = (9 × 8 × 7) / (3 × 2 × 1) = 84.

Therefore, there are 84 ways to distribute the 10 pencils so that each child gets at least one.