Mississippi

Palindrome- Word that can be read the same forwards as backwards

MISSISSIPPI is a 11 letter word that contains 1M, 4I, 4S, 2P

Because the number of letters is odd and there is an odd number of M's, the M must go in the middle.

_ _ _ _ _ M _ _ _ _ _

After that, the 5 letters on either side of the word will be a mirror image of each other. Thus, it is only necessary to consider one side of the word in the calculation as the other side is fixed once the letters on the left are chosen.

So there are 5 spots and 5 letters, two of which repeat twice (S and I), yielding the following calculation:

$\frac {5!}{2! \times 2! \times 1!}$ = 30

Thus there are 30 possible ways to rearrange the word MISSISSIPPI such that each word is palindromic.

You have 1M, 2p, 4i, 4s. take 1 m in middle and both side you have 1p, 2s, 2i. You can do this in $\frac{5!}{2! * 2!}$