In how many ways can we distribute $11$ pieces of candy among $4$ kids, provided that every kid gets at least $1$ piece of candy?
I know how to solve this problem one way, but why does this way not work:
Number of ways to distribute to $4$ people $-$ number of ways to distribute to $4$ people such that at least one person doesn't get any candy
number of ways to distribute to $4$ people (just normal stars and bars): $^{14}C_3$
number of ways to distribute to $4$ people such that at least one person doesn't get any candy: $4\times ^{13}C_2$. This is because we can "merge" two of the bars, and then multiply by $4$ because there are $4$ people whose bars we can "merge".
$^{14}C_3 - 4(^{13}C_2) = 52$.
The answer is $120$.
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