Track / Stage 3 / 240 of 260 #720 of 2444

Problem 720

AMC 10/12, early questions

Combinatorics Difficulty 3.9 Find the answer PMO Area Stage · Philippines

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.

Official 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.

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