Question

How many different ways are there of arranging the letters in the word ACCOMMODATION if no two Cs may be together?

Answer 1

`109771200`

Explanation:

ACCOMMODATION has `13` letters comprising:

• `3` O's
• `2` each of A, C, M
• `1` each of D, T, I, N

If the letters were all different, then there would be `13!` ways of arranging them.

As it is, the total number of distinct ways of arranging all `13` letters is:

`(13!)/(3!2!2!2!) = 6227020800/(6*2*2*2) = 6227020800/48 = 129729600`

If the two letter C's are adjacent, then it is as if we are arranging `12` objects, with:

• `3` O's
• `2` each of A, M
• `1` each of D, T, I, N and CC

The number of ways that we can do that is:

`(12!)/(3!2!2!) = 479001600/(6*2*2) = 479001600/24 = 19958400`

So the total number of ways of arranging the letters of ACCOMMODATION with no `2` C's together is:

`129729600 - 19958400 = 109771200`