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## Homework Statement

There is a group of 7 people. How many groups of 3 people can be made from the 7 when 2 of the people refuse to be in the same group?

## Homework Equations

_{n}C

_{r}

## The Attempt at a Solution

Here is what I know:

_{7}C

_{3}gives the total number of groups that can be made.

_{5}C

_{1}*

_{2}C

_{2}gives the number of groups with the two feuding people; because,

_{2}C

_{2}gives the number of ways to choose both of the feuding people, and

_{5}C

_{1}gives the number of ways to choose any 1 of the none feuding people. Their product gives the number of sets with the two feuding people and 1 of the group of non-feuding people.

Based on this

_{7}C

_{3}-

_{5}C

_{1}*

_{2}C

_{2}gives the correct answer.

I understand this is the easiest method to solve the problem; however, I don't understand why

_{5}C

_{2}*

_{2}C

_{1}doesn't give the number of groups that do not contain the feuding people.

Can anyone explain what I am missing, conceptually?

I understand that (

_{7}C

_{3}-

_{5}C

_{1}*

_{2}C

_{2})≠(

_{5}C

_{2}*

_{2}C

_{1}). But

_{5}C

_{2}gives the number of ways to choose 2 people from the set of non-feuding people.

_{2}C

_{1}gives the number of ways to choose only 1 of the set of feuding people, right/wrong? I thought the product of these two combinations would give the number of sets that do not contain the feuding people, but I am wrong.

_{5}C

_{2}*

_{2}C

_{1}gives fewer than the actual number of sets that do not contain the feuding people. I gotta be missing something conceptually, what is it?