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June 5th, 2016, 11:11 PM   #1
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a problem related to the counting principle

Suppose that you want to make license plates that consist of three letters followed by three digits. The letters can be chosen from A to Z and the digits can be chosen from 0 to 9.

You are allowed to use the same letter twice and the same number twice. How many license plates can you make?

my attempt

26x26x25x10x10x9=15210000

I multiply 26 by itself twice because you can repeat one letter. After you have repeated one letter, you have only 25 letters to choose from. The same reason applies to the three digits.

My friend told me that my answer is wrong. Can someone please explain the problem? Thanks a lot.
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June 6th, 2016, 12:30 AM   #2
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If I'm not mistaken, the answer is this:

(26P3 + 26 * 25 * 3C2) * (10P3 + 10 * 9 * 3C2) = 17 374 500

Because under your count, either one of these two is not counted:

A B A 1 1 2
B A A 1 1 2

(By analogy, you also didn't count one of BAB112/ABB112, etc.)

which leads to under-counting.
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June 6th, 2016, 12:40 AM   #3
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If the first 2 letters are A and B, the 3rd choice can be any of the 26, it only drops to 25 when the other 2 letters are the same, which only happens 26 times (AA, BB etc....)
The same would be true for the numbers.
I suggest the answer may be:
((26x26x26)-26) x ((10x10x10)-10) = 16,731,000

Edit:
Having re-read the question, it it stating that you can't have all 3 letters or numbers the same, AAA, BBB etc, which only happens 26 and 10 times respectively.
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Last edited by weirddave; June 6th, 2016 at 12:45 AM.
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June 6th, 2016, 02:09 AM   #4
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Quote:
Originally Posted by weirddave View Post
If the first 2 letters are A and B, the 3rd choice can be any of the 26, it only drops to 25 when the other 2 letters are the same, which only happens 26 times (AA, BB etc....)
The same would be true for the numbers.
I suggest the answer may be:
((26x26x26)-26) x ((10x10x10)-10) = 16,731,000

Edit:
Having re-read the question, it it stating that you can't have all 3 letters or numbers the same, AAA, BBB etc, which only happens 26 and 10 times respectively.
I think you may have made a calculation error. I input yours in my calculator, but I got 17374500, same as my answer...
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June 6th, 2016, 05:07 AM   #5
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Quote:
Originally Posted by 123qwerty View Post
I think you may have made a calculation error. I input yours in my calculator, but I got 17374500, same as my answer...
That's quite odd, I must have hit another button at some point in the calculation, ta
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