### The letters of the word STATISTICAL are rearranged in a random order. What is the probability that the letters C and L have exactly 3 letters between them?

 14 110

Step by Step Explanation:
1. The first step is finding out all the possible arrangements of the letters of the word STATISTICAL.
Note that we are not interested in "unique" arrangements (in some kind of problems we would be), but just the total possible arrangements.
2. There are 11 letters in the word STATISTICAL. So, they can be arranged in 11! ways i.e. 11 × 10 × 9 × ... 2 × 1 ways.
To see this, if you pick any letter, it can be placed in any of the possible 11 positions.
Now, the second letter can be placed in any of the other 10 positions (11 minus the one taken up by the first letter).
So, the two letters can be arranged in 11 × 10 ways. Similarly we can arrange the remaining letters.
3. Now, we know that the total number of possible arrangements = 11!
We need to look for the possible arrangements where the letters C and L have exactly 3 letters between them.
4. This can be seen by inspection.
If C is at the first position (first letter of the word), then L will be at position 5 (since there are 3 letters between them).
Similarly, if C is at the second position, then L will be at position 6.
There are 7 such possible arrangements, the last one having C at the 7th position and L at the last position.
5. Interchanging the positions of L and C in all the cases given in the above step, we get 7 possible arrangements.
So, we can place the letters C and L in 14 ways.
Now, we have filled in 2 of the 11 positions with C and L in 14 ways.
Since there is no restriction on the remaining 9 letters, the number of possible arrangements of 9 letters in 9 positions is 9!. So, the total ways to rearrange all the letters so that C and L have exactly 3 letters between them is 14 × 9!.
6. Now, we can work out the probability of rearranging the letters of the word STATISTICAL such that C and L have exactly 3 letters between them.
P (arrangement) =
 Number of possible arrangements where C and L have 3 letters between them Total number of possible arrangements of the letters of the word STATISTICAL
=
 14 × 9! 11!
=
 14 110