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

Sep 19, 2016

$109771200$

#### 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:

$129729600 - 19958400 = 109771200$