2006 AMC 10A Problem 18

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

Answer: C

Concepts:arrangements with restrictions multiplication principle

Difficulty rating: 1450

Small hint:

Glue the two letters into a single block that moves as a unit

Big hint:

Count digit choices, letter choices, and positions for the block

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.