Question

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

Answer 1

Hint: Here in this question, we have to find the different ways of arrangement of the letters in the word ACCOMMODATION, to solve this we use the permutation concept. Here first we determine the number of ways of the word ACCOMMODATION and we find the number of ways where the two C’s are together and we subtract it.

Complete step-by-step answer:
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.
Now here in this question we have to find the different ways of arranging the letters in the word ACCOMMODATION where the two C’s should not be together.
In the word ACCOMODATION we have 13 letters. In the 13 letters the letter C is present two times, O is present 3 times and A is present 2 times and M is present 2 times.
Therefore the arrangement of the letters in the word ACCOMMODATION is given as
 \[\dfrac{{13!}}{{3!2!2!2!}}\]
On simplifying this we have
 \[ \Rightarrow \dfrac{{13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3!}}{{3! \times 2 \times 1 \times 2 \times 1 \times 2 \times 1}}\]
On cancelling the terms we get
 \[ \Rightarrow 13 \times 6 \times 11 \times 5 \times 9 \times 4 \times 7 \times 6 \times 5 \times 4\]
On multiplying these numbers we get
 \[129729600\]
Therefore there are 129729600 ways where we arrange the word.
Now we consider the two C’s as one letter and then we find the different ways. so we have
 \[\dfrac{{12!}}{{3!2!2!}}\]
On simplifying this we have
 \[ \Rightarrow \dfrac{{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3!}}{{3! \times 2 \times 1 \times 2 \times 1}}\]
On cancelling the terms we get
 \[ \Rightarrow 6 \times 11 \times 5 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4\]
On multiplying these numbers we get
 \[19958400\]
Therefore there are 19958400 ways where we arrange the word such that the two C’s are together.
Now we have to find the number of ways of arranging the letters in the word ACCOMMODATION such that the two C’s are not together
so we have \[129729600 - 19958400\]
On simplifying we get
 \[ \Rightarrow 109771200\]
Therefore there are 109771200 ways where we arrange the word such that the two C’s are not together.
So, the correct answer is “109771200”.

Note: Here in this question there is no direct method where we apply to get the solution. So first we find the number of ways the letters are arranged in the word. and then we find the number of ways the letters are arranged in the word such that the two C’s are together. Here we must know the permutation concept and the formula \[n! = n \times n - 1 \times ... \times 2 \times 1\]