Answer :
There are [tex]10[/tex] letters in the word [tex]CALCULATOR[/tex].
Start by pretending that all the letters are different, then the total arrangements would be simply [tex]10![/tex]
But, there are [tex]2[/tex] C's, [tex]2[/tex] A's, and [tex]2[/tex] L's.
For each of these, there will be repititions, as swapping two same letters wont produce a different arrangement. There are [tex]2![/tex] ways to swap C's, [tex]2![/tex] ways to swap A's, and [tex]2![/tex] ways to swap L's. Dividing these duplicates gives the answer :
[tex]\dfrac{10!}{2!{\cdot}2!{\cdot}2!}[/tex]
Start by pretending that all the letters are different, then the total arrangements would be simply [tex]10![/tex]
But, there are [tex]2[/tex] C's, [tex]2[/tex] A's, and [tex]2[/tex] L's.
For each of these, there will be repititions, as swapping two same letters wont produce a different arrangement. There are [tex]2![/tex] ways to swap C's, [tex]2![/tex] ways to swap A's, and [tex]2![/tex] ways to swap L's. Dividing these duplicates gives the answer :
[tex]\dfrac{10!}{2!{\cdot}2!{\cdot}2!}[/tex]