# Number of combinations of donuts with restrictions and repetitions involved

Ten people visit a donut shop that sells the following donut flavors: Mint, Chocolate, Strawberry, and Sugar.

Jack is allergic to Sugar flavor while David, Brock and Charles are allergic to Strawberry flavor. How many different combinations of ten donuts can the group choose (with repetition)?

Hi guys, I have been struggling with this question for the past few hours and have still not been able to solve it. Could anyone help me out or suggest a solution? Thanks in advance!!

• What are your thoughts? Sure, this question is rather sweet... – Parcly Taxel Oct 31 '16 at 10:15
• If each person chooses one donut, then the question is asking how many solutions there are in the nonnegative integers to the equation $$x_1 + x_2 + x_3 + x_4 = 10$$ subject to the restrictions that $x_3 \leq 7$ and that $x_4 \leq 9$. Alas, the question does not make clear that each person is selecting one donut. – N. F. Taussig Oct 31 '16 at 10:16
• @N.F.Taussig: And if each person gets the doughnut that he/she chooses, then the answer is simply $4^3\cdot6^4$. – barak manos Oct 31 '16 at 10:40

I think that J, B, C, D each has 3 flavors to choose, which is $3^4$.
The other 6 people each has 4 flavors to choose, which is $4^6$.
By the General Product Principle, $$3^4×4^6$$ is the final answer.