What is the probability that a randomly selected permutation of the letters A, A, C, C, L, L, O, R, T, U would spell "calculator"?

See explanation

Step-by-step explanation:

Total number of letters = n! = 10

Permutation for 10 letters = n!

                                          = 10!

                                          = 10*9*8*7*6*5*4*3*2*1

                                          = 3628800

From n letters:

Number of A = 2

Number of C = 2

Number of  L = 2

A₁, A₂, C₁, C₂, L₁, L₂, O, R, T, U

Out of 3628800 permutations the word calculator can be spelled as:

C₁ A₁ L₁ C₂ U L₂ A₂ T O R

C₂ A₂ L₂ C₁ U L₁ A₁ T O R

and so on

So total number of different permutations are  

(permutation of number of letters) / permutation of (number of As * number of Cs * number of Ls)

  =  10! / (2! 2! 2 !)  

  = 10*9*8*7*6*5*4*3*2*1 / (2*1) (2*1) (2*1)

  = 3628800 / 2 * 2 * 2

  =  3628800 / 8

  =   453600

Hence probability that a randomly selected permutation of the letters A, A, C, C, L, L, O, R, T, U would spell "calculator" is:

 1 / total number of different permutations  

= 1 / 10! / (2! 2! 2 !)  

= 1/453600

= 0.000002