Combination of at most one of the oldest 2 players

Question

11 players must be selected from 15 players. (i) Find at most one of the oldest two players must be included. Please help with full annotation, thnx

Two cases: (i) None of oldest two is included or (ii) exactly one is. Or else: Count the number of unrestricted choices and subtract the bad ones (both oldest are included). The number of bad choices is $\binom{13}{9}$. — André Nicolas, Mar 17 '14 at 06:57

score 1 · Accepted Answer · answered Mar 20 '14 at 23:23

Obviously, 15 players are composed of the 2 oldest and 13 younger players.

Ways to select 1 of the oldest and 10 of the younger: ${2 \choose 1}{13 \choose 10}$

Ways to select none of the oldests and 11 of the younger: ${2\choose 0}{13 \choose 11} = {13 \choose 11}$

Sum them to select at most one of the oldests among 11 selected players: ${2 \choose 1}{13 \choose 10}+{13 \choose 11}$

$${2 \choose 1}{13 \choose 10}+{13 \choose 11} = \frac{2!13!}{10!3!}+\frac{13!}{11!2!}=650$$

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