Question

Find the number of distinct permutations of the word "CALCULATOR."​

Answer 1

Given:

The word is "CALCULATOR."​

To find:

The number of distinct permutations.

Solution:

In the word "CALCULATOR",

The number of letters = 10

In which, we have 2 C, A, L and 1 U, T, O, R.

We know that,

$^nC_r=\dfrac{n!}{(n-r)!}$

$^nC_n=\dfrac{n!}{(n-n)!}=n!$

To find the number of distinct permutations of the given word need to divide permutation of all letter by the permutations of repeated letters.

$\text{Total ways}=\dfrac{^{10}C_{10}}{^2C_2\ ^2C_2\ ^2C_2\ ^1C_1\ ^1C_1\ ^1C_1\ ^1C_1}$

$\text{Total ways}=\dfrac{10!}{2!2!2!1!1!1!1!}$

$\text{Total ways}=\dfrac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 2\times 2\times 1\times 1\times 1\times 1}$

$\text{Total ways}=453600$

Therefore, the number of distinct permutations of the word "CALCULATOR" is 453600.