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### Number and Letter Combinations

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Date: 07/29/97 at 20:25:38
Subject: Possible number combinations

I have a question concerning number calculations. How many possible
combinations can a person make with letters (lower case) a-z
and 1-9; starting from 1 digit and ending up with up to 8 digits?
For example  az3 or 1d or ae4rg or 5634f45.

How did you figure it out - what formula did you use?

Thanks.
```

```
Date: 07/30/97 at 05:50:19
From: Doctor Anthony
Subject: Re: Possible number combinations

First consider the number of combinations when no repetition of
letters or numbers is allowed.

I will use the notation nCr to represent the number of combinations
of r things that can be formed from n things.  For example, 9C4 is the
number of combinations of 4 things that can be made from 9 things

9 x 8 x 7 x 6
The value of 9C4 is  --------------- = 126
1 x 2 x 3 x 4

9!
It can also be calculated from   ------
4! 5!

There are 26 letters and 9 numbers giving 35 things from which to
choose.  These can then be chosen in combinations ranging in size from
1 to 8 in the following number of ways:

35C1 + 35C2 + 35C3 + 35C4 + 35C5 + 35C6 + 35C7 + 35C8

= 35 + 595 + 6545 + 52360 + 324632 + 1623160 + 6724520 + 23535820

= 32267667

This is the answer when no repetition of letters or numbers is
allowed.

If repetitions are allowed then the calculation is simpler, since we
can choose every position of a combination of size, say 3, in 35 ways,
giving 35^3 combinations. For combinations of size 1 to 8, the total
is

35 + 35^2 + 35^3 + 35^4 + 35^5 + 35^6 + 35^7 + 35^8

= 2.31810702...  x 10^12

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/

```

```
Date: 07/31/97 at 00:18:27
From: Ann
Subject: Re: Possible number combinations

Thank you very much!
```
Associated Topics:
High School Permutations and Combinations

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