Seven little men live in a little cottage.
Their names are:
Outside are seven little chairs all in a row...
How many different ways can they sit in the chairs so that no one sits next to someone whose name starts with the same letter?
e.g. Dopey can't sit next to Doc.
Note: Dwarves that have the same first initial are distinguishable, so, for example, if Dopey and Doc trade places, it's a different arrangement.
More permutations problems
Image credit: www.disneyclips.com
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There are 7 ! ways that the seven men can occupy the seven chairs.
2 ⋅ 6 ! of them have Dopey sitting next to Doc.
This is because if we treat Dopey and Doc as one unit sitting together, they have 2 ways to sit next to each other, and then 6! ways that they can sit down with the other dwarves.
Similarly for Sneezy and Sleepy...
2 ⋅ 6 ! of the combinations have Sneezy sitting next to Sleepy.
However if we subtract off both of the above, we have double counted by 4 ⋅ 5 !
This comes from the fact that if we look at how many combinations there are for Sneezy AND Sleepy and Dopey and Doc to both be sitting together, we can treat each as a unit, giving 5 ! ways they can sit, but for each of them Sneezy could be on the right of Sleepy and vice versa, and same with Dopey and Doc, thus giving the factor of 4 .
Therefore, the answer is:
7 ! − 2 ⋅ 6 ! − 2 ⋅ 6 ! + 4 ⋅ 5 ! = 2 6 4 0