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

##### 1 Answer

Sep 19, 2016

#### Explanation:

ACCOMMODATION has

#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

As it is, the total number of distinct ways of arranging all

#(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

#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

#129729600 - 19958400 = 109771200#