Candy bar combinations

Question

A local mart sells 6 kinds of candy bar. You want to buy 15 candy bars. How many possibilities are there if you want more than 5 bars of any one of the kinds?

Is there any hint to solve this questions?

Answer 1

This problem can be solved by using Stars and Bars combinatorial theorems. See also extended stars-and-bars problem(where the upper limit of the variable is bounded)

Let $x_i$ be the number of bars of kind $i$ with $i=1,\dots,6$. Note that there are at most $2$ kinds of candy bar with more than $5$ bars (because $3\cdot 6>15$).

Case 1: $\binom{6}1$ ways to choose a kind and, by Stars and Bars, $\binom{14}{5}$ non-negative integer solutions of $$y_1+x_2+x_3+x_4+x_5+x_6=15-6=9$$ with $y_1=x_1-6$.

Case 2: $\binom{6}2$ ways to choose the two kinds and, by Stars and Bars, $\binom{8}{5}$ non-negative integer solutions of $$y_1+y_2+x_3+x_4+x_5+x_6=15-6-6=3$$ with $y_1=x_1-6$, $y_2=x_2-6$.

Hence the final answer is $$\binom{6}1\binom{14}{5}-\binom{6}2\binom{8}{5}.$$