How Many Ways Can You Arrange 3 Birds to Feed if Two Don't Get Along?

[Happyzor]

Homework Statement

You have 7 birds lined up to feed. Only 3 birds can feed at a time. Two of the 7 birds do not like to feed with each other. How many combinaions can be formed?

Homework Equations

C(n,k)=n!/k!(n-k)!

The Attempt at a Solution

Attempted solution
C(7,3)-C(7,2)=21.

That is the wrong answer. The correct answer is C(7,3)-C(2,2)*C(5,1)=5

Can someone explain this to me? Thanks. My thinking was 7 choose 3 minus 7 choose the two birds that don't like to mix. Subtract and you'll get the birds that mix. Obviously its wrong -_-. Thanks for your help in advance.

 

[VeeEight]

C(7,3) is clear.
Then you must subtract the number of times when the two birds are with each other. You must fix these two birds - that is where C(2,2) comes from (ie, these is only 1 way to select these two birds). The C(5,1) term comes from the fact that after you have selected these two birds, you must pick 1 last one so you have 3 on the feeding line.

Also, note that if 3 is the maximum number of birds allowed on the feeding line, then you can still consider situations when there are just 1 and 2 birds (respectively) on the line.

 

[Happyzor]

C(7,3) is clear.
Then you must subtract the number of times when the two birds are with each other. You must fix these two birds - that is where C(2,2) comes from (ie, these is only 1 way to select these two birds). The C(5,1) term comes from the fact that after you have selected these two birds, you must pick 1 last one so you have 3 on the feeding line.

Also, note that if 3 is the maximum number of birds allowed on the feeding line, then you can still consider situations when there are just 1 and 2 birds (respectively) on the line.

Ah ok, thanks. I was thinking order mattered and all, but now its clear order doesn't matter. There is only one way to choose the two birds. Then the 3rd combination is the combinations with the other 5 birds. Thanks a bunch for clearing it up.