SOLUTION: find the number of ways of arranging the 26 letters in the english alphabet in a row such that there are exactly 5 letters between x and y?

The x and y can be from this way
where x comes 1st and y comes 7th

x-----y-------------------

to this case where x comes 20th
and y comes 26th:

-------------------x-----y

So that's 20 ways to place the x 6 places before the y.

And there are 20 more ways to place the x 6 places after the y

y-----x-------------------

to

-------------------y-----x

That's 40 ways to place the x and y.

For each of those 40 ways to place the x and y, the remaining 
24 letters can be arranged in 24! ways.

Answer: (40)(24!) = 24817936069329577574400000

Edwin