How many sevendigit telephone numbers have an odd number of

How many seven-digit telephone numbers have an odd number of even numbers?

Solution

We assume that all strings of 7 digits are allowed.

Take any string of 6 digits. If it has an odd number of even digits, it can be completed to a number with an odd number of even digits by appending any of the 5 odd digits.

If it has an even number of even digits, it can be completed to a number with an odd number of even digits by appending any of the 5 even digits.

So any of the 10⁶ 6-digit strings can be completed in 5 ways to a 7-digit string with an odd number of even digits.

It follows that the total number is 5×10⁶.