# Combinations of people on a team

Sample question for a test. Not sure how to approach:

A mutant volleyball team of 3 heros and 3 villains is to be chosen from a group of 9 heros and 5 villains.

1. How many different teams are possible?

2. How many different teams are possible if two of the heros refuse to be on the same team?

3. How many different teams are possible if two of the villains refuse to be on the same team?

4. How many different teams are possible if one hero and one villain refuse to be on the same team?

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For reference: Combinations

1. $9 \choose 3$ $5 \choose 3$

2. Since we already have the total number of teams from part 1, it's easiest to see which teams are now illegal. Put the two disagreeable heroes together and observe there are 7 heroes that can go in the third slot, so we now have (our answer from 1) - 7$5 \choose 3$

3. Same reasoning as 2.

4. Similar to 2, put the disagreeable hero and villain together, and we end up having to subtract $8 \choose 2$ $4 \choose 2$

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Is it not ${8 \choose 2}{4 \choose 2}$ for the last part – 00000000000001 Jan 23 '12 at 7:21
Yes, it is, sorry. Fixing that now... – Chad Miller Jan 23 '12 at 7:23
Well, at least I understand it! – 00000000000001 Jan 23 '12 at 7:27