the number of permutations of all the letters of the word STATISTICIAN

Question

Determine the number of permutations of all the letters of the word STATISTICIAN if the arrangement must begin with C and end in N.

my work:After we fix first letter as S and last letter as N we have 10 letters left

TATISTITIA

Here we have 4 Ts, 2 As ,3 Is, and 1 A.

If all letters were different, the result would have been 10!.

To account for repetitions we need to divide this by (4!*2!*3!).

So, we get 10!/(4!*2!*3!)=12600.

is this the correct way to solve ?

Answer 1

If it begins with $C$ and ends with $N$, then the remaining letters are:

STATISTIIA

That is $\{S\cdot 2, T\cdot 3, A\cdot 2, I\cdot 3\}$

So, we are looking for the number of permutations of this multiset, which is:

$$\dfrac{10!}{2!3!2!3!} = 25200$$

Your answer in the comments is correct. Your problem with the first time you did it is that you fixed the S and the N, rather than the C and the N.