College Algebra, Chapter 10, Review Exercises, Section Review Exercises, Problem 38

In the Zip+4 postal code system, zip codes consists of nine digits.

a.) How many different Zip+4 codes are possible?

There are ten numbers from 0-9. If we assume that repetition is allowed, then the possible numbers of Zip+4 codes is

$10^9 = 1,000,000,000$

b.) How many different Zip+4 codes are palindromes? (A palindrome is a number that reads the same from left to right as right to left.)

A 9-digit palindrome is the same as an 8-digit palindrome with one extra digit in the middle, which can be filled by any if the 10 numbers from 0-9. For an 8-digit palindrome, there are 4 spaces in which you can use 4 different numbers to form an 8 digit palindrome. You can place the first 4 in any order and the last 4 in reverse order. Therefore, the number of 8-digit palindrome is

$10 \times 10 \times 10 \times 10 = 10,000$

Then, for a 9-digit palindrome, we have

$10 \times 10 \times 10 \times 10 \times 10 = 100,000$

c.) What is the probability that a randomly chosen Zip + 4 code is a palindrome?

If we have $100,000$ palindrome numbers out of $1,000,000,000$ numbers, then the probability in this case is

$\displaystyle \frac{100,000}{1,000,000,000} = 0.001$