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Problem 827

AMC 12 late, AIME early

Combinatorics Difficulty 4.5 Multiple choice CEMC Fermat · Canada · 2019

There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. How many different-looking arrangements are possible?

Pick one

Official solution

Since 4 balls are chosen from 6 red balls and 3 green balls, then the 4 balls could include:

4 red balls, or
3 red balls and 1 green ball, or
2 red balls and 2 green balls, or
1 red ball and 3 green balls.

There is only 1 different-looking way to arrange 4 red balls.

There are 4 different-looking ways to arrange 3 red balls and 1 green ball: the green ball can be in the 1st, 2nd, 3rd, or 4th position.

There are 6 different-looking ways to arrange 2 red balls and 2 green balls: the red balls can be in the 1st/2nd, 1st/3rd, 1st/4th, 2nd/3rd, 2nd/4th, or 3rd/4th positions.

There are 4 different-looking ways to arrange 1 red ball and 3 green balls: the red ball can be in the 1st, 2nd, 3rd, or 4th position.

In total, there are $1+4+6+4=15$ different-looking arrangements.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.