Group of 6

Probability Level 3

There are $4$ adults, $8$ boys and $12$ girls in a camp. They want to create $4$ groups each consisting of $1$ adult, $2$ boys and $3$ girls. How many ways are there to create teams of $6$ this way?

24640 442368 931392000 136048896 22353408000

1 solution

Saad Khondoker
Feb 23, 2021

First assign adults to teams. There is only 1 way to do it as the teams can not be differentiated (they don't have names or individuality).
Then we assign boys to each team. Number of ways to do it is $^{8}$ $C_{2}$ $*$ $^{6}$ $C_{2}$ $*$ $^{4}$ $C_{2}$ $= 2520$ .
Then we assign girls to each team. Number of ways to do it is $^{12}$ $C_{3}$ $*$ $^{9}$ $C_{3}$ $*$ $^{6}$ $C_{3}$ $= 369600$ .
Total number of ways to assign $= 1 * 2520 * 369600 = 931392000$

The question asks "how many ways are there to create teams of 6?" Not "how many ways are there to divide the group into four teams of six?"

Richard Desper - 3 months, 2 weeks ago

I thought I made it clear in the previous sentence. And even if you got confused, I didn't give any option where you make every team of 6 not just the specific kind with 1A2B3G. But thanks for pointing it out.

Saad Khondoker - 3 months, 2 weeks ago

Group of 6

1 solution

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