SOLUTION: how many different teams of 5 children can be chosen from a group of 7 boys and 11 girls if each team must have at least 2 boys on it

First we'll find the number of teams of 5 from the 18 children without
restrictions of gender.  Then we'll subtract the number with no boys
and the number with just 1 boy.

1.  The number of teams with no restriction of gender.

That's 18C5 = 8568 teams with no restrictions of gender.

2.  The number of teams with no boys, which means 5 girls:

That's 11C5 = 462 teams of all girls, no boys, which we must subtract.

3.  The number of teams with exactly 1 boy and 4 girls:

We can choose the 1 boy 7C1 = 7 ways.
Then we can choose the 4 girls 11C4 = 330 ways.
Thats 7(330) = 2310 ways, which we also must subtract.

Final answer = 8568-462-2310 = 5796 teams with at least 2 boys.

Edwin