Question

) if the letters of the word regulations be arrange at random, what is thechance that there will be exactly 4 letters between the r and the e?

Answer 1

Alright, there are 11 distinct alphabets in the word REGULATIONS. If we fix the letters r and e at 1st and 5th places and arrange the remaining letters we get exactly 4 letters between r and e. and we can arrange them in 9! ways.

Like wise, we can fix r and e at 6 places namely, 1st and 5th
2nd and 6th
3rd and 7th
 4th and 8th
5th and 9th
6th and 10th
7th and 11th

So the number of ways of arranging the letters according to your question would turns out to be 6 x 9! ways. And total number of ways of arranging them is 11! ways.
  Probability= 6x9!÷11!⇒3/55 is the chance of arranging the words according to the question.