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June 5th, 2016, 11:11 PM  #1 
Senior Member Joined: Apr 2008 Posts: 194 Thanks: 3  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. 
June 6th, 2016, 12:30 AM  #2 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics 
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 undercounting. 
June 6th, 2016, 12:40 AM  #3 
Senior Member Joined: Apr 2014 From: UK Posts: 953 Thanks: 340 
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 reread 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. Last edited by weirddave; June 6th, 2016 at 12:45 AM. 
June 6th, 2016, 02:09 AM  #4  
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics  Quote:
 
June 6th, 2016, 05:07 AM  #5 
Senior Member Joined: Apr 2014 From: UK Posts: 953 Thanks: 340  

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