Permutations?

[megchan]

I'm working on an online math homework, and basically...I am confused.

"How many distinguishable horizontal ar-
rangements of all the letters in ALABAMA
are possible?"

I did 7! and put the answer 5,040 and ofcourse its wrong.
I'm struggling with how to solve these problems when there are repeats.

 

[galactus]

There are 7 letters in ALABAMA.

But, 4 A's. Divide by the factorial of the number of repeats. In this case, only A's repeat.

So, the number of arrangements is \(\displaystyle \frac{7!}{4!}\).

Say we wanted the number of arrangements of MISSISSIPPI.

There are 11 letters. But, there are 4 S's, 4 I's, 2 P's.

\(\displaystyle \frac{11!}{4!4!2!}\).

See now?.

 

[megchan]

There are 7 letters in ALABAMA.

But, 4 A's. Divide by the factorial of the number of repeats. In this case, only A's repeat.

So, the number of arrangements is \(\displaystyle \frac{7!}{4!}\).

Say we wanted the number of arrangements of MISSISSIPPI.

There are 11 letters. But, there are 4 S's, 4 I's, 2 P's.

\(\displaystyle \frac{11!}{4!4!2!}\).

See now?.

Thank you!