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Permutation License plate problem

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Permutation License plate problem

by dzelkas » Sun Sep 16, 2007 7:12 pm
Can someone help me explain the answer in this problem?

A license plate consists of a combination of 6digits or letters. All numbers (0-9) and all 26 letters may be used. How many Unique license plates are there?

a) 36^6
b) 36!/30!6!
c)36!/30!
d)36!/6!
e)30!

Correct answer is A)

I am confused because the way I thought this problem should of been done was 36 total unique characters. and we need to make a unique combo of of 6 so I thought C) was the right answer.

Any help?
thanks

Danny
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by mschling52 » Mon Sep 17, 2007 8:18 am
Answer C would be correct if we were looking for the number of different ways to arrange the 36 characters with the requirement that each be unique. It comes from the permuation formula:

(n P r) = (n!)/(n-r)!

which gives the number of ways to different ways to pick r objects from a set of n, where the order of the r objects selected matters. So, this would be correct if we were asked how many license plates would be possible if each letter/number could only appear once on each plate.

However, in this question, there is nothing to state that the license plate AAAAAA could not be used. Therefore, there are 36 choices for the first character, 36 for the second, and so on. The total number of license plates is then 36*36*36*36*36*36.

Hope that helps answer your question.
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by wongee » Sun Sep 23, 2007 12:11 pm
But the question says that the license plate is a COMBINATION of letters and numbers. Doesnt that mean that the AAAAAA option isnt really an option?
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by wongee » Sat Sep 29, 2007 12:20 pm
Thanks for the explanation, hopefully I wont choose to go with my reasoning on the actual day.
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by Leonard C » Mon Oct 01, 2007 2:58 am
Guys,

The key here is to note that the nPr and nCr formulas only apply when there is replacement. (That is why the factorial term is used).

Example 1: How many license plate combinations are there if you can use all 36 letters and nos, but you can only use one letter or number once and the order of the letter and nos matters?

Here, because there is no replacement (ie you can only use one letter or number once) and because the order matters you must use nPr.

Example 2: How many license plate combinations are there if you can use all 36 letters and nos, but you can only use one letter or number once and the order of the letter and nos does not matter?

Here, use nCr.

Example 3: How many license plate combinations are there if you can use all 36 letters and nos, order matters, and you can use letters and nos twice?

Here you us 36^n.
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by beatthegmat » Mon Oct 01, 2007 4:52 pm
Moved this thread to the PS section...
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