Number of ways of distributing n identical things among r persons
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In this example, there are n = 12 n=12 n=12 identical objects to be distributed among r = 9 r=9 r=9 distinct bins. Using the above formula, the number of possible distributions is ( 11 8 ) = 165 \binom{11}{8}=\boxed{165} (811)=165. You are at a party with 4 of your friends (5 distinct people in total).
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