How many of permutation of letters in the word "$\bf{MISSISSIPPI}$" are palindromes?
$\text{My Try}:$ Palindrome is a word which read as same forwards and backwards.
Here word "$\bf{MISSISSIPPI}$" contain $\bf{4I\;,4S\;,2P}$ and $\bf{1M}$. So total $11$ letters.
So must be in the form of $—–M—–$,
so we put $\bf{2I\;,2S\;,1P}$ on left of $M$,
which can be done as $\displaystyle =\frac{5!}{2!\times 2!\times 1} = 30$.
But I did not understand how can we not permute $\bf{2I\;,2S\;,1P}$ on right of $M$?
Best Answer
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 $30$.