Question

how many different arrangements are possible from the letters of the word CALCULATOR

Answer 1

There are $10$ letters in the word $CALCULATOR$.
Start by pretending that all the letters are different, then the total arrangements would be simply $10!$

But, there are $2$ C's,  $2$ A's, and $2$ L's.
For each of these, there will be repititions, as swapping two same letters wont produce a different arrangement. There are $2!$ ways to swap C's,  $2!$ ways to swap A's,  and $2!$ ways to swap L's. Dividing these duplicates gives the answer :
$\dfrac{10!}{2!{\cdot}2!{\cdot}2!}$