Question

By using the letters of the word “RESPONSIBLE”, find the number of ways to arrange these letters if there are no restrictions.

Answer 1

"Responsible" has 11 letters, but there are two e's and two s's. So, the number of permutations is

$\dfrac{11!}{2!2!}=\dfrac{11!}{4}$

In fact, there are $11!$ permutations of 11 elements. But if you have repeating elements, you must divide this by the factorial of the number of occurences of each repeating letter. In this case, we have two letters that appear twice, so we divide by $2!$ twice.