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

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- Dec 31st 2008, 05:34 AMmathaddictpermutation
Find the number of words formed by permuting all the letters of the word EXERCISES .

- Dec 31st 2008, 05:48 AMMush
- Dec 31st 2008, 05:51 AMIsomorphism
- Dec 31st 2008, 06:23 AMSoroban
Hello, mathaddict!

This is a permutation with some duplicated objects.

There is a formula for this, but you can derive it yourself.

Quote:

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

If we had 9letters, there would be $\displaystyle 9!$ possible words.__different__

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: .$\displaystyle \frac{9!}{3!\,2!} \:=\:30,\!240$