SOLUTION: How many distinguishable ways can be written using all the letters in the word ALGEBRA?

Question 879135: How many distinguishable ways can be written using all the letters in the word ALGEBRA?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
Number of letters in this word are:
A = 2
L = 1
G = 1
E = 1
B = 1
R = 1
There are 7 total letters with 2 of them being the same.
The formula is 7! / 2! = 2520.

This is hard to show because there are so many possibilities, but I can show it with a simpler type problem.

Assume the letters ABC.
The number of possible arrangements are 3! = 6
Those arrangements are:
ABC
ACB
BAC
BCA
CAB
CBA

Now assume the letters AAC.
The number of possible arrangements are 3! / 2! = 3.
Those arrangements are:
AAC
ACA
CAA

You have essentially replace the B with another A.
Where you had ABC and ACB, you now only have AAC and ACA
Where you had BAC and BCA, you now only have AAC and ACA
Where you had CAB and CBA, you now only have CAA and CAA.
You now have AAC twice and ACA twice and CAA twice.
The number of unique arrangements is only 3 which are AAC and ACA and CAA.

The general formula for number of unique permutations is:

P = n! / (x1! * x2! * ... xn!) where:

n is the number of letters.
x1, x2, ..., xn are the number of letters that are the same.

For example:

Original letters are ABCDEFG

Formula is 7!

Original letters are AACCEEG

Formula is 7! / (2! * 2! * 2!)

Original letters are AAAAABB

Formula is 7! / (5! * 2!)

That's how it works.

RELATED QUESTIONS
How many distinguishable ways can the letters of the word STATISTICS be written ? (answered by Fombitz)

In how many distinguishable ways can the letters in the word STATISTICS be arranged?... (answered by ikleyn)

In how many distinguishable ways can letters of the word TENNESSEE be... (answered by sudhanshu_kmr)

how many distinguishable ways can the letters of the word VERTICAL be arranged using only (answered by checkley77)

How many words can be written using all the letters of the word... (answered by ikleyn)

how many distinguishable permutations can be made from the letters in the word... (answered by edjones)

How many distinguishable permutations can be made of the letters in the word STEGOSAURUS? (answered by ikleyn)

How many distinguishable 6 letter words can be formed using the letters in... (answered by Edwin McCravy)

How many distinguishable "words" can be formed from the letter of the word COLLEGES? (all (answered by stanbon)