Find the number of words formed by permuting all the letters of the word EXERCISES .
Hello, mathaddict!
This is a permutation with some duplicated objects.
There is a formula for this, but you can derive it yourself.
Find the number of words formed by permuting all the letters of the word EXERCISES.
If we had 9 different letters, there would be $\displaystyle 9!$ possible words.
But there are 3 identical E's.
Switching them would not produce a new word.
So our answer is too large by a factor of $\displaystyle 3!$
. . (The number of ways 3 objects can be arranged.)
And there are 2 identical S's.
They can be switched in $\displaystyle 2!$ ways without producing a new word.
So our answer is also too large by a factor of $\displaystyle 2!$
So the answer is: .$\displaystyle \frac{9!}{3!\,2!} \:=\:30,\!240$