Mississippi

How many different words can we form from the letters of the word MISSISSIPPI?

Solution
We can choose a position for the letter M in \(\binom{11}{1}\) ways. We can choose positions for four letter I's in the remaining 10 positions in \(\binom{10}{4}\) ways. Then we can choose positions for four letter S's in the remaining 6 positions in \(\binom{6}{4}\) ways. The remaining two positions are for the letter P.

Another approach leading to the same result: Pretend that the letters in a word are indistguishable at first and then we will gradually distinguish them, e.g. with colors – four distinct shades of blue for I, four distinct shades of red for S, etc. We can color the four uncolored I's with four shades of blue in \(4!\) ways. Similarly the four S's also give \(4!\) colorings and we can color the two P's in two ways. (We need not color the letter M, which is itself distinguished from the other letters.) If we try all possible colorings of all uncolored words, we obtain all possible arrangements of 11 colored letters, of which there are \(11!\).
Answer
The number of possible words is \( \frac{11!}{4!\cdot 4!\cdot 2!} =34650\).