Question

How do you find the number of distinct arrangements of the letters in INSISTS?

Answer 1

`(7!)/(2!3!) = 420`

Explanation:

If all of the `7` letters were distinct then we could arrange them in `7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040` different ways.

We can simulate that by giving the repeated letters subscripts...

`"I"_1 "N" "S"_1 "I"_2 "S"_2 "T" "S"_3`

For each possible arrangement, there are `2! = 2 * 1 = 2` different possible arrangements of the `"I"`'s and `3! = 3 * 2 * 1 = 6` different possible arrangements of the `"S"`'s.

Hence the number of distinct arrangements of:

`"INSISTS"`

is:

`(7!)/(2!3!) = 5040/(2*6) = 420`