Delegation with at least one girl

Text

A class has $18$ students: $10$ girls and $8$ boys. We want to form a delegation of $4$ students containing at least one girl. In how many ways?

Solution

We count by complement. Delegations of any $4$ students number $\binom{18}{4} = \frac{18\cdot 17\cdot 16\cdot 15}{24} = 3060.$ Those with no girl (all boys) number $\binom{8}{4} = \frac{8\cdot 7\cdot 6\cdot 5}{24} = 70.$ Hence the delegations with at least one girl number $N = 3060 - 70 = \boxed{2990}.$