An engineering group consists of 12 men and 13 women.
If 2 women refuse to be on the same team together, how many different project teams can be formed consisting of 5 men and 5 women?
An engineering group consists of 12 men and 13 women.
If 2 women refuse to be on the same team together, how many different project teams can be formed consisting of 5 men and 5 women?
Count the number of ways to choose $5$ men from the $12$ available; I think that you know how to do that. Now call the awkward women $A$ and $B$.
Alternatively, you can count the number of unusable $5$-woman groups, i.e., those that contain both $A$ and $B$, and subtract that from the number of all possible $5$-woman groups.
Now you have the total number of ways in which you can choose the women. Multiply by the number of ways to choose the men, and you’re done.
Hint: for the women, you can select 5 of the 11 who will work with anybody, or select 4 of 11 plus 1 of 2.