Competition problem — Combinatorics

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Combinatorics Difficulty 3.9 AMC 10/12 Find the answer Philippines

Problem:

How many ways can you place $10$ identical balls in $3$ baskets of different colors if it is possible for a basket to be empty?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

This is a problem of distributing $10$ identical balls into $3$ distinct baskets, where baskets may be empty.

The number of ways is given by the stars and bars formula:
$\binom{10 + 3 - 1}{3 - 1} = \binom{12}{2} = 66.$

So, there are $66$ ways.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.