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

Question

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

Is the answer 1680??

Answer 1

Yes, the answer is 1,680: a and b have 14 distinct position pairings that have 2 or 3 letters in between (because they could be in 1st and 4th, 2nd and 5th, ..., 1st and 5th, 2nd and 6th ... and in either order). Once we have counted those 14 arrangements for a and b, the remaining 5 letters can be placed in any order in the remaining 5 slots, which gives 5P5 = 120 possible arrangements for c - g. And 14 x 120 = 1,680.