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### Combinations of Letters

```
Date: 10/20/98 at 03:04:55
From: Anthony
Subject: General formula for permutations and combinations

To Dr. Math:

I have been set a piece of work involving finding a general rule for
the different number of arrangements that can be made out of people's
names, such as EMMA (12 combinations) and LUCY (24).

I have come up with this formula:

Number of letters!
---------------------------
Number of different letters

e.g.:

6!
Hannah =  --------
2!x2!x2! (because it has 2 h's, 2 n's, and 2 a's)

10!
Commitment = -----
3!x2! (because it has 3 m's and 2 t's)

What I need your help on is how to put this formula into a general
term. I have found the top line (N!) but I am having difficulty with
the bottom one. Could you please show me how you deduce the formula?

Yours in anticipation,
Anthony
```

```
Date: 10/20/98 at 10:48:55
From: Doctor Rob
Subject: Re: General formula for permutations and combinations

Count the frequencies of the letters that appear. In COMMITMENT,

C -> 1
O -> 1
M -> 3
I -> 1
T -> 2
N -> 1
E -> 1

Note that the sum of these numbers must be the total number of
letters:

1 + 1 + 3 + 1 + 2 + 1 + 1 = 10

Then the numerator is the factorial of the sum, and the denominator is
the product of the factorials of the counts:

10!/(1!*1!*3!*1!*2!*1!*1!) = 10!/(3!*2!) = 302400

since 1! = 1. (If you like, you can throw in the other letters, like Q
and Z, with frequency counts 0. They don't change the sum since
10 + 0 = 10, nor the product of the factorials of the counts, since
0! = 1.)

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

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