How many permutations of the letters in the word MISSISSIPPI are palindromes?

We permute the letters on the right in the way which will make the $11$ letter word a palindrome.

For example, we begin with $${\bf-----M-----}.$$ There are $30$ ways of permuting the two $\bf{I}$'s, two $\bf{S}$'s, and one $\bf{P}$ that we must use on the left. Suppose we choose the string $\bf{IPSSI}$ so that we have $${\bf IPSSIM-----}.$$ As we want the word to be a palindrome, we need to use the string $\bf{ISSPI}$ on the right, which is just the string we used on the left but reversed. In this case we get $${\bf IPSSIMISSPI}.$$

In general, we don't need to consider the permutations of letters on the right because the string is fixed once the string on the left is chosen, so the number of palindromes is just the number of choices for the string of letters on the left, which is $30$.

Once you put the letters to the left of $M$ then the letters to the right are fixed. They are the letters to the left in the reverse order

You figured out the placement of the M nicely. Once we know that, we need only consider the left side since once we place a letter on the left side, the same letter must be placed on the corresponding place on the right side of M.

Then there are $5$ spots with $5$ possible letters to choose from, that is $5!$ total combinations. However, both the I and S repeat (since we have $2$ of them to place on the left side of M), to avoid repetitions, we must have $$ \frac{5!}{2! \cdot 2!}=\frac{120}{4}=30 $$ So there are $30$ palindromes.