My school's math club has 6 boys and 8 girls.  I need to select a team to send to the state math competition.  We want 6 people on the team.  In how many ways can I select the team to have more girls than boys?
We do this problem with three cases.

Case 1: 4 girls, 2 boys on the team.

With 4 girls on the team, there are $\binom{8}{4} = 70$ ways to pick the girls, and $\binom{6}{2} = 15$ ways to pick the boys, for a total of $70 \times 15 = 1050$.

Case 2: 5 girls, 1 boy on the team.  With 5 girls on the team, there are $\binom{8}{5} = 56$ ways to pick the girls, and $\binom{6}{1} = 6$ ways to pick the boy, for a total of $56 \times 6 = 336$.

Case 3: 6 girls on the team. With 6 girls on the team, there are $\binom{8}{6} = 28$ ways to pick the girls on the team.

This gives us a sum of $1050 + 336 + 28 = \boxed{1414}$.