Math Help - permutation

1. permutation

Find the number of words formed by permuting all the letters of the word EXERCISES .

Find the number of words formed by permuting all the letters of the word EXERCISES .
$n ! = n \times (n-1) \times (n-2) \times (n-3) \times ... \times 1$ where $n$ is the number of letters in the word.

Assuming that by "words" you mean combinations of letters, rather than coherent words with literary meanings.

Find the number of words formed by permuting all the letters of the word EXERCISES .
Do you know that the number of arrangements of objects where $a_1$ are of one kind, $a_2$ are of another kind,... $a_n$ are of the nth kind is $\frac{(a_1 + a_2 + ... a_n)!}{a_1 ! a_2 ! ... a_n!}$

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 $9!$ 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 $3!$

. . (The number of ways 3 objects can be arranged.)

And there are 2 identical S's.
They can be switched in $2!$ ways without producing a new word.
So our answer is also too large by a factor of $2!$

So the answer is: . $\frac{9!}{3!\,2!} \:=\:30,\!240$