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Hannisch
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Homework Statement
How many "words" with 5 letters can be created from the letters in the word ALGEBRA? Each letter can be used only once.
Homework Equations
The Attempt at a Solution
I know the answer to this (1320) and I know how I got to it (which I'll describe in a minute), but I know that while my way of getting it is right, it's. not the easiest, and I can't seem to figure out how on Earth they want me to do it.
I know that choosing 5 things from a set with 7 elements can be done in
[itex]\frac{7!}{(7-5)!}[/itex]
ways. I also know that if I were to use all the letters of ALGEBRA the answer would be
[itex]\frac{7!}{2!}[/itex]
since one letter is used twice. In this case these happen to be exactly same, but I can't seem to put these principles together to get a correct answer. Help, please?
The way I solved it was more of a brute force solution. The correct answer is 11 * 5!, and I got to that conclusion by reasoning that 5! of the words are made by the letters LGEBR, which have no recurring letters. Then I "exchanged" one of the letters from that to an A, one at a time, and then to both As. This can be done in 11 ways:
LGEBR
AGEBR
LAEBR
LGABR
LGEAR
LGEBA
AAEBR
LAABR
LGAAR
LGEAA
AGEBA
And all of them can be chosen in 5! different combinations. I know this is the correct answer (it's an online based homework, I've inputted this and it says I'm right).