Question

A witness to a hit-and-run accident tells the police that the license plate of the car in the accident, which contains three letters followed by three digits, starts with the letters\({\rm{AS}}\)and contains both the digits\({\rm{1 and 2}}\). How many different license plates can fit this description?

Short answer

There are \(1560\) different license plates.

Step-by-step solution

Introduction

The product rule is a formula in calculus that is used to compute the derivatives of two or more functions.

Explanation

There are \({\rm{26}}\) letters and \({\rm{10}}\) digits

First letter: \({\rm{1}}\) way (since it has to be A)

Second letter: \({\rm{1}}\)way (since it has to be S)

Third letters: \({\rm{26}}\)ways

First digit: \({\rm{1}}\) way (since it has to be\({\rm{1}}\) )

Second digit: \({\rm{1}}\) way (since it has to be\({\rm{2}}\))

Third digit: \({\rm{10}}\)ways

Order three digits: \({\rm{P(3,3)}}\) (since the order of the positions is important and

repetition not allowed)

Use the product rule:

\(\begin{array}{c}{\rm{1 \times 1 \times 26 \times 1 \times 1 \times 10 \times P(3,3)}}\\{\rm{ = 26 \times 10 \times 3!}}\\{\rm{ = 1560}}\end{array}\)

Hence, there are \(1560\) different license plates.