Problem Solving

In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

The answer is 25401600.

Can anybody explain me how?

First, determine the number of ways to fill out the PXXXXS unit. This unit is 6 letters long: _ _ _ _ _ _. First determine the possibilities for the first and last letters. We have 2 choices for the first letter and 1 for the last letter: 2 _ _ _ _ 1. Now, there are ten remaining letters that could go in the blanks, so 2*10*9*8*7*1 total ways to determine this unit. Now, treat the unit as one and arrange it with the remaining 6 letters which could be done in 7! ways. Multiply these together to get 2*10*9*8*7*1*7!. Finally the letter T appears twice in PERMUTATIONS, so divide all of this by 2 to get 10*9*8*7*7! or [spoiler]7*10!=25401600[/spoiler]

PERMUTATIONS

P/S _ _ _ _ P/S _ _ _ _ _ _ => 10!/2 * 2
_ P/S _ _ _ _ P/S _ _ _ _ _ => 10!/2 * 2
_ _ P/S _ _ _ _ P/S _ _ _ _ => 10!/2 * 2
_ _ _ P/S _ _ _ _ P/S _ _ _ => 10!/2 * 2
_ _ _ _ P/S _ _ _ _ P/S _ _ => 10!/2 * 2
_ _ _ _ _ P/S _ _ _ _ P/S _ => 10!/2 * 2
_ _ _ _ _ _ P/S _ _ _ _ P/S => 10!/2 * 2

Total no of ways = 7 * 10!/2 * 2 = 25401600