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2006 AMC 10A Problem 18

Problem 18 of 25IntermediateCombinatorics

A license plate in a certain state consists of 44 digits, not necessarily distinct, and 22 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?

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Solution

Since the two letters must be adjacent, treat them as one block. A plate is then 44 digits plus this block—55 objects—and the block can occupy 55 positions. There are 10410^4 choices for the digits and 26226^2 for the two letters, so the total is 5⋅104⋅262.5 \cdot 10^4 \cdot 26^2. Thus, the correct answer is C.
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Tagged: arrangements with restrictions · multiplication principle

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