Permutations of the letters a, b, c, d, e, f, g

[Caldus]

How many permutations of the letters a, b, c, d, e, f, g have either two or three letters between a and b?

My guess for this is that if a and b have to have two letters between them, then there are 5! ways to arrange the rest of the letters, right? Same deal if a and b have to have three letters between them. So:

5! + 5!

Would be my answer. Am I anywhere close with this one? Thank you.

 

[mathman]

Your answer is correct for each specific (a,b) position pair. Your have to multiply by all possible position pairs for a and b, e.g. (1,4), (4,1), (2,5) etc. for two spacing, and similarly for three spacing.

 

[Caldus]

Thank you.

I came up with this new answer:

(5! * 4) + (5! * 3) = 840

Am I close here?

 

[mathman]

You're missing a factor of 2, since a at 1 and b at 4 is different from b at 1 and a at 4, etc., unless you are assuming a is always before b. In that case you are right.