At a table in restaurant six people ordered roast beef three

At a table in restaurant six people ordered roast beef, three ordered turkey,two ordered pork chops, and one ordered flounder. of course no two portions of any of these items are absolutely identical the 12 servings are brought from the kitchen

in how many ways can they be distributed so that everyone get the correct item?

in how many ways can they be distributed so that no one can get the correct item?

Solution

1.

Even if you permute the 6 roast beefs, that\'s still okay. There are 6! = 720 ways to do that.

Even if you permute the 3 turkeys, that\'s still okay. There are 3! = 6 ways to do that.

Even if you permute the 2 pork chops, that\'s still okay. There are 2! = 2 ways to do that.

The flounder order must get to that person.

Thus, there are 720*6*2 = 8640 ways to get the correct items. [ANSWER, 8640]

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2.

The turkeys, pork chops, and flounder must all go to those who ordered roast beef. Otherwise, someone gets it correctly.

There are 6! = 720 ways to do that.

The 6 pork chops can also be permuted, and there are 6! = 720 ways to do that.

Hence, there are 720*720 = 518400 ways to get them all wrong. [ANSWER]