n how many ways can the letters of the word PERMUTATIONS be arranged if the there are always 4 letters between P and S? Choose one.
A) 25401600	B) 25401800	C) 254016200	D) 26401600	E) 27401600
There are 12 words in letter PERMUTATIONS. Out of which T is repeated twice.   Now first we need to see how many ways we can make word with 4 letter between P and S.   Except P and S there are total of 10 letters, so number of way of selecting them = 10C4 = 210   Also note that question is asking to place exactly 4 words between P and S, but does not tells you if P has to be the first letter of S has to be the first letter. So In all the above combinations, we can rotate the position of P and S.   So total way = 210*2= 
<gadget id="calculator">210*2</gadget>
<output>420</output>
420   The selected 4 letters can be rotated between P and S in= 4! ways   So total ways = 420 * 4!   Consider this 6 letter chunk (P, S, and 4 letter between them) as 1 letter.   Remaining letters are 6. So in total we have 7 letters, which can be arranged in 7! ways.   So total number of ways = 7! * 420 * 4!   Now since letter T was repeated twice, we should divide the above result by 2!.   So Total number of ways = 7! * 420 * 4! / 2! = 25401600 25401600
<result>A</result>
n how many ways can the letters of the word PERMUTATIONS be arranged if the there are always 4 letters between P and S?