Question

Our school's girls volleyball team has 14 players, including a set of 3 triplets: Missy, Lauren, and Liz. In how many ways can we choose 6 starters if the only restriction is that not all 3 triplets can be in the starting lineup?

Answer 1

There are 11 players who can be chosen at will
If no triplet is chosen, you have 11C6 ways to do it
If 1 is chosen, you have 11C5*3C1 ways to do it
If 2 are chosen, you have 11C4*3C2 ways to do it
so add them up

Answer 2

If all triplets are in the starting lineup, we are choosing the 3 remaining starters from 11 players, which can be done in $\binom{11}{3} = \boxed{165}$ ways.

Answer 3

answer is 165

Answer 4

According to AoPS, the answer is 2838

Answer 5

990 according to AoPs Alcumus

Answer 6

2838