Question

In Exercises 5–36, express all probabilities as fractions.

Classic Counting Problem A classic counting problem is to determine the number of different ways that the letters of “Mississippi” can be arranged. Find that number.

Short answer

The number of different (unique) ways in which the letters of the word “Mississippi” can be arranged is 34650.

Step-by-step solution

Given information

The letters of the word “Mississippi” have to be arranged.

Permutation in the case of repetition of letters

Several different permutations/arrangements can be made for n units if $n_1$ are of one type, $n_2$ are of another type, and so on.

$\frac{n!}{n_1!\times n_2!\times ...}$

Compute the number of ways different arrangements for the letters

The total number of letters (n) is 11.

The number of times the letter “i” is repeated is 4.

The number of times the letter “s” is repeated is 4.

The number of times the letter “p” is repeated is 2.

The total number of permutations possible to arrange the letters of the word “Mississippi” is given as follows:

$$\begin{gathered} \frac{11!}{4!\times 4!\times 2!}=\frac{11\times 10\times 9\times 8\times 7\times 6\times 5\times 4!}{4!\times 4\times 3\times 2\times 1\times 2\times 1} \\ =34650 \end{gathered}$$

Therefore, the number of different ways in which the letters of the word “Mississippi” can be arranged is 34650.