2006 AMC 10A Problem 18

Problem 18 of 25IntermediateCounting & Probability

A license plate in a certain state consists of

4

digits, not necessarily distinct, and

2

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

4

digits plus this block—

5

objects—and the block can occupy

5

positions. There are

10^4

choices for the digits and

26^2

for the two letters, so the total is

5 \cdot 10^4 \cdot 26^2.

Thus, the correct answer is C.

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