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

Answer:

#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#