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### Combinations of Letters and Numbers

```
Date: 3/19/96 at 10:22:57
From: Mike Hansen
Subject: Math Contests

I just participated in a Math Contest. I missed 5th place by 1

A moped license plate has two letters and then four numbers in it.
How many plates can be made without duplicating?

Mike Hansen
```

```
Date: 3/20/96 at 1:35:8
From: Doctor Patrick
Subject: Re: Math Contests

Hi Mike!  Is that  without duplicating numbers and letters, or
plates?

Since I'm not sure, I'll show you how to do both.

The easier one is to find is the second, so let's start with
that. The first digit is a letter, so you have 26 choices.   Each
of those 26 possible choices for the first digits can be followed
by 26 more choices for the second digit.  So to find the total
number of letter combinations we multiply the number of choices
for the first digit by the number of choices for the next digit,
in this case 26*26, for a total of 676.

Do you see how that works?

Then we do the ame for the numbers. There are four digits and
each of them can be from 0 to 9, for 10 possible choices per
digit. That gives us 10*10*10*10, 10000 combinations.

Now we just need to combine the two parts.  There were 676
possible letter combinations  multiplied by 10000 number
combinations for each of them.  I'll let you calculate the final
step on your own, using the methods from above.

In the case of no duplicate digits we must multiply the 26
possible first digits by 25 (26 letters - the one already used.
This gives us 26*25 letters, which equals 650.  Then we move on to
the numbers.  There are 10 for the first digit, then 9 for the
second (10 -1 for the number already used).

How many do you think will be in the third and fourth digits?

Once you have those four numbers, multiply them out as we did
above, and then put the two parts (letters and numbers) together.

Good luck!

-Doctor Patrick,  The Math Forum
```

```
Date: 3/20/96 at 9:54:1
From: Mike Hansen
Subject: Re: Math Contests

I got:

26(letters) * 26(letters) * 10(digits) * 10(digits) * 10(digits) *
10(digits) = 6,760,000

That is what they had on the answer sheet, but they changed it to
three million something. My teacher and I do not know how they
came up with this answer. My way is perfectly logical.
```

```
Date: 3/20/96 at 11:42:24
From: Doctor Patrick
Subject: Re: Math Contests

Was the answer they gave 3,276,000?  If so, then they wanted you
to do the problem in the second way I gave.  This was the number
of plates that could be found with no letters or numbers repeated
inside of the same plate.

26 (letters) * 25 (the unused letters) *10 (numbers) * 9 (unused
numbers) ...

Do you think you can finish this on your own?  First find the
number of remaining choices for the last two digits, then multiply
it out.

Hope this helps,

-Doctor Patrick,  The Math Forum

```
Associated Topics:
High School Permutations and Combinations

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