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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?

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6 answers
  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

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  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.

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  3. answer is 165

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  4. According to AoPS, the answer is 2838

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  5. 990 according to AoPs Alcumus

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  6. 2838

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