Question

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?

$10^4 \cdot 26^2$

$10^3 \cdot 26^3$

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

$10^2 \cdot 26^4$

$5 \cdot 10^3 \cdot 26^3$

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.

2006 AMC 10A Problem 18

Solution:

Problem 18 in Other Years