Answer to Question #146450 in Combinatorics | Number Theory for Josh

Question

Find the number of distinguishable permutations that can be formed from the letters of the word "CINCINNATI"
a.604800
b.25200
c.50400
d.100800

Answer 1

$CINCINNATI: 2C, 3I, 3N, 1A,1T$

There are 10 letters in the word "CINCINNATI"

The number of permutations of 10 elements is given by the following formula:

$$P_{10}=10!$$

The letter $C$ is repeated 2 times in the word.

The letter $I$ is repeated 3 times in the word.

The letter $N$ is repeated 3 times in the word.

The number of distinguishable permutations that can be formed from the letters of the word "CINCINNATI" is

$$\dfrac{10!}{2!\cdot3!\cdot3!}=\dfrac{10(9)(8)(7)(6)(5)(4)(3)(2)(1)}{2\cdot6\cdot6}$$

$$=\dfrac{3628800}{72}=50400$$

c. 50400