Dopey and Doc?

Seven little men live in a little cottage.

Their names are:

  • Dopey
  • Sneezy
  • Bashful
  • Doc
  • Sleepy
  • Grumpy
  • Happy

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


The answer is 2640.

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2 solutions

Geoff Pilling
Jan 15, 2018

There are 7 ! 7! ways that the seven men can occupy the seven chairs.

2 6 ! 2 \cdot 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 ! 2 \cdot 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 ! 4 \cdot 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 ! 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 4 .

Therefore, the answer is:

7 ! 2 6 ! 2 6 ! + 4 5 ! = 2640 7! - 2 \cdot 6! - 2 \cdot 6! + 4 \cdot 5! = \boxed{2640}

Number of ways 7 dwarves can arrange themselves w.r.t. 7 chairs = 7!

Combine Dopey and Doc and call them D. Also combine Sleepy and Sneezy and call them S. Now we have to deal with 5 ! multiplied by the positions they may occupy together as a group and positions they may interchange amongst themselves within the combined group.

Number of ways in which D and S will then sit be = 5! x 20.

So, the number of ways the same initial letter named dwarves will never sit next to each other: 7! - (5! x 20) = 2640.

Interesting approach... Can you explain where the 20 comes from? And how do you deal with the cases where say, D+D sit together but not S+S, and vice versa?

Geoff Pilling - 3 years, 4 months ago

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It was a laborious task of counting cases. 20 in total. As per your question, 2 Ds and 2 Ss can never sit together. So when you consider the case of combining them and subtracting them from the general 7! case, the case you have mentioned about Ds being together and Ss not being together or vice versa, automatically gets eliminated.

A Former Brilliant Member - 3 years, 4 months ago

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