Phone Numbers

Problem created by Tao Steven Zheng (郑涛)

Problem

Suppose every telephone number consists of seven digits such that the first digit of all telephone numbers is an odd number, and the last digit of all telephone numbers is a non-zero even number. Question: What is the maximum number of seven-digit telephone numbers that can be issued?

A rotary dial phone

Solution

Each telephone number has seven digits, so imagine there are 7 cells. The first digit must be odd; thus, there are only 5 choices for the first cell: 1, 3, 5, 7, or 9. The last digit must be a non-zero even number; thus, there are only 4 choices for the first cell: 2, 4, 6, or 8. The the remaining 5 cells can be filled by any digit from 0 to 9; hence, there are 10 choices each.

Subsequently, the maximum number of telephone numbers is

$$\begin{align} 5 \times 10 \times 10 \times 10 \times 10 \times 10 \times 4 &= 5 \times 4 \times {10}^{5} \\ &= 2000000 \\ &= 2\times {10}^{6} \end{align}$$