# How many ways to pair wine and food? [closed]

I have the following homework, for which I'd like you to give me some hints. Thank you.

There are M food dishes and N bottles of different wines. There are 4 ways to pair food and wine: 1 (dish) to 1 (type of wine), 1 to many, many to 1, and many to many. A menu must use all M dishes and N wine bottles. How many possible menus can one make?

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## closed as off-topic by Fundamental, Semiclassical, TooTone, RecklessReckoner, Mike MillerAug 10 '14 at 23:15

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Fundamental, Semiclassical, TooTone, RecklessReckoner, Mike Miller
If this question can be reworded to fit the rules in the help center, please edit the question.

(a) The problem might look familiar when you think of "food dishes" as a sequence of $M$ blanks to fill in and "wines" as $N$ distinct digits $0, 1, \ldots, N-1$. For example, what does the question ask (in these terms) when $N=10$? (b) You need to decide how to interpret the statement. Is it four different questions or one? –  whuber Feb 28 '11 at 22:29
This is combinatorics rather than probability or statistics, so it might be better at math.SE –  onestop Mar 1 '11 at 10:18

## 1 Answer

I don't understand "There are 4 ways to pair food and wine: 1 (dish) to 1 (type of wine), 1 to many, many to 1, and many to many" If you have a number of courses, you need that many dishes and that many wines. The naive answer is that you can pair n wines with the first dish, n-1 with the next, and so on, leading to n! choices. But that doesn't respect the 4 ways to pair food and wine. What am I missing?

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