Q:

how many x contain atleast one y

Description
A group of 5 animals is to be chosen from 6 cats and 4 dogs.
Question
how many groups contain at least one dog?
Working Out
There are at most 4 dogs, so a group of 1 dog and 4 cats meet the condition as does a group of 2 dogs and 3 cats, 3 dogs, 2 cats and 4 dogs 1 cat.
So all I need to do is add the combinations together like so C(5,4) + C(5,3) + C(5,2) + (5,1) which equals 30 I believe.
Does that seem correct to you fokes? I have a feeling its wrong.

A:

Choose 5 animals out of (6 + 4) animals. This can be done in $C(10, 5)$ ways. Now calculate the ways in which groups can be formed where the group members are only cats. This can be done in $C(6,5)$ ways.
So $C(10, 5) - C(6,5)$ are the ways in which a group can have atleast one dog.

