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,
[tex]^nC_r=\dfrac{n!}{(n-r)!}[/tex]
[tex]^nC_n=\dfrac{n!}{(n-n)!}=n![/tex]
To find the number of distinct permutations of the given word need to divide permutation of all letter by the permutations of repeated letters.
[tex]\text{Total ways}=\dfrac{^{10}C_{10}}{^2C_2\ ^2C_2\ ^2C_2\ ^1C_1\ ^1C_1\ ^1C_1\ ^1C_1}[/tex]
[tex]\text{Total ways}=\dfrac{10!}{2!2!2!1!1!1!1!}[/tex]
[tex]\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}[/tex]
[tex]\text{Total ways}=453600[/tex]
Therefore, the number of distinct permutations of the word "CALCULATOR" is 453600.
Number of distinct permutation of the word "CALCULATOR." is 453600
How to solve this problem?
Total number of words in "CALCULATOR" is 10
we find out the words that are repeating
A is repeating twice (2)
C is repeating twice (2)
L is repeating twice (2)
Number of permutations = number of letters !/ repeating letters!
Number of distinct permutation =
[tex]\frac{10!}{2!2!2!} \\\frac{10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{2\cdot2\cdot2} \\453600[/tex]
Number of distinct permutation of the word "CALCULATOR." is 453600
learn more about the distinct permutations here:
https://brainly.com/question/21767354
