[Math] How many different words can be formed using all the letters of the word GOOGOLPLEX

combinatoricsmultinomial-coefficientspermutationssolution-verification

How many different words can be formed using all the letters of the word GOOGOLPLEX?

I tried answering this problem
and came up with the formula $n!/a!b!c!$
where $n$ in this case is 10-because it is the number of letters in the word.

$a!$ is O and in this case the letter O is repeated 3 times so $a=3$.

$b!$ is G and in this case the letter G is repeated 2 times.

and $c!$ is the other letters which are not repeated.

So the answer that i got is 302,400

Am I correct?

Best Answer

Your answer is off by a factor of $2$, seems that you forgot the double letter L.

The word GOOGOLPLEX consists of the letters O (3x), G (2x), L (2x), P (1x), E (1x) and X (1x). So the number of possible words consisting of these letters is the multinomial coefficient

$$ \binom{10}{3,2,2,1,1,1} = \frac{10!}{3!\cdot 2!\cdot 2!\cdot 1!\cdot 1!\cdot 1!} = 151200. $$

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