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/