In how many ways can we arrange the 13 letters of the word “COMMUNICATION” in which
(i} there are no restriction.
(ii) the word start with M and end with I.
(iii) the two letters C do not occur next to each other.
Here's the third one :
Let's count the ways where the two C's can appear together
They can appear together in any one of twelve positions
And the number of identifiable arrangements of the other letters is
11 ! / [ 2! * 2! * 2! * 2! ]
So.....the total number of ways where they can occur together is
12 * 11! / [ 16] = 29,937,600
So.....the number of ways in which they don't appear together is
194,594,400 - 29,937,600 = 164,656,800