Question

Motorcycle plates in colorado have three numbers followed by three letters. how many motorcycle plates area available with these restrictions? (assume there are no other restrictions.)

Answer 1

When you have a counting problem like this, you have to think in terms of how many choices you have. A generic plate is like this:

$x_1x_2x_3x_4x_5x_6,\quad x_1,x_2,x_3 \in \{0,1,\ldots,9\},\quad x_4,x_5,x_6 \in \{a,b,\ldots,z\}$

so, you have 10 choices for each of the first three places, and 26 choices for each of the last three places.

This leads to a total of

$10 \cdot 10 \cdot 10 \cdot 26 \cdot 26 \cdot 26 = 10^3 \cdot 26^3 = (10 \cdot 26)^3 = 260^3 = 17576000$

possible plates.