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I am having problem with the question below,

How many words can be formed out of the letters of the word "courage"?

As the author did not mentioned about repetition so I answer should be 7^7 but the real answer is 7!. Please help me out why is it so.


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I think it's implied that each letter can only be used once. But did the author specify 7-letter words? – Will Dana Oct 16 '11 at 4:11
@WilliamD.: No, he did not. – Fahad Uddin Oct 16 '11 at 4:15
I would support the claim that there is no clear answer. If you want 7 letter words but letters can be duplicated you are right. If you want 7 letter words without duplicates, 7! is correct. If you want words without duplication, but shorter than 7 is acceptable you need a sum. – Ross Millikan Oct 16 '11 at 4:25
up vote 2 down vote accepted

This is a permutation problem if no letter is to be repeated. So you have $7$ letters and you want to form words with all $7$, so you should have $\frac{7!}{(7-7)!}=7!$. Alternatively, note that the first position could be filled in $7$ ways since you have $7$ letters, the second position $6$ ways since you have only $6$ letters now and so on. So you have $7!$ words you can form.

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