# Combinatorial question: How many ways to eat lunch?

1,015

## Solution 1

You have $4$ entrees to pick from, and you choose 1. $$4\choose1$$ You have 8 side dishes to pick from, and you choose 2. $$8\choose2$$ Put those together to get $$\binom{4}{1}\binom{8}{2}=4\cdot28=112$$

Keep in mind that this assumes the two side dishes must be different. If they can be the same, then there are $8\choose1$ ways to pick a doubled up sidedish. You can add this to the original $8\choose2$ for all your side dish combinations, then multiply by your entree possibilities for a new answer.

## Solution 2

If she must choose exactly $1$ entree and exactly $2$ unique side-dishes, then:

• Choose $1$ out of $4$ items: $\binom{4}{1}=\frac{4!}{1!\times3!}=4$
• Choose $2$ out of $8$ items: $\binom{8}{2}=\frac{8!}{2!\times6!}=28$
• So she can make $4\times28=112$ different combinations
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### laura

Updated on August 08, 2022