Combinatorics - Choosing group memebers

[adam.wp]

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

_nC_r

The Attempt at a Solution

Here is what I know:

₇C₃ gives the total number of groups that can be made.

₅C₁*₂C₂ gives the number of groups with the two feuding people; because, ₂C₂ gives the number of ways to choose both of the feuding people, and ₅C₁ 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 ₇C₃-₅C₁*₂C₂ gives the correct answer.

I understand this is the easiest method to solve the problem; however, I don't understand why ₅C₂*₂C₁ 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 (₇C₃-₅C₁*₂C₂)≠(₅C₂*₂C₁). But ₅C₂ gives the number of ways to choose 2 people from the set of non-feuding people. ₂C₁ 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. ₅C₂*₂C₁ gives fewer than the actual number of sets that do not contain the feuding people. I got to be missing something conceptually, what is it?

 

[Dick]

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

_nC_r

The Attempt at a Solution

Here is what I know:

₇C₃ gives the total number of groups that can be made.

₅C₁*₂C₂ gives the number of groups with the two feuding people; because, ₂C₂ gives the number of ways to choose both of the feuding people, and ₅C₁ 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 ₇C₃-₅C₁*₂C₂ gives the correct answer.

I understand this is the easiest method to solve the problem; however, I don't understand why ₅C₂*₂C₁ 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 (₇C₃-₅C₁*₂C₂)≠(₅C₂*₂C₁). But ₅C₂ gives the number of ways to choose 2 people from the set of non-feuding people. ₂C₁ 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. ₅C₂*₂C₁ gives fewer than the actual number of sets that do not contain the feuding people. I got to be missing something conceptually, what is it?

5C2*2C1 gives you the number of groups that contain exactly one feuding person. It doesn't count the groups that don't have either one.

 

[adam.wp]

Awesome! I forgot all about the sets with neither. So:

₅C₂*₂C₁+₅C₃ gives the answer I was looking for. Since ₅C₃ is the number of sets which contain neither person and ₅C₂*₂C₁ are the sets with either.

So I needed to think about which groups contained either, and which contained neither.

Thank you!