Chapter 15.5, Problem 13WE

To determine

To calculate: The number of license plates of 3 symbols with at least two letters.

Answer to Problem 13WE

The number of license plates of 3 symbols with at least two lettersare $37856$ .

Explanation of Solution

Given information:

The license plates of 3 symbols that include both letters and symbols with at least two letters is to be formed.

Formula used:

Fundamental counting principle states that if there are m ways to make a selection and n ways to make a second selection then there are $m\times n$ number of ways in which two selection can be made.

Calculation:

Consider the provided information that license plates of 3 symbols that include both letters and symbols with at least two letters is to be formed.

There are 26 letters and 10 digits from 0 to 9.

Case I: When there are exactly two letters.

Let 3-symbol license platebe $\boxed{}\boxed{}\boxed{}$ .

First two boxes have 26 possibilities each as 26 letters are there and the last box has 10 ways in which it can be filled.

  $\boxed{26}\boxed{26}\boxed{10}$

Recall that Fundamental counting principle states that if there are m ways to make a selection and n ways to make a second selection then there are $m\times n$ number of ways in which two selection can be made.

Therefore, the number of ways to select are,

  $26\times 26\times 10=6760$

But as the digit can be at any place among the 3 places in the license plate so number of possible 2-letter plates are,

  $6760\times 3=20280$

Case I: When there are all 3 letters.

Let 3-symbol license plate be $\boxed{}\boxed{}\boxed{}$ .

All the boxes have 26 possibilities each as 26 letters are there.

  $\boxed{26}\boxed{26}\boxed{26}$

Recall that Fundamental counting principle states that if there are m ways to make a selection and n ways to make a second selection then there are $m\times n$ number of ways in which two selection can be made.

Therefore, the number of ways to select are,

  $26\times 26\times 26=17576$

There are 17576 3 letter license plates.

There total number of plates is the addition of the above mutually exclusive possibilities.

  $17576+20280=37856$

Thus, number of license plates of 3 symbols with at least two lettersare $37856$ .