Chapter 13, Problem 17RE

Telephone Numbers Using the digits 0, 1, 2,...,9, how many 7-digit numbers can be formed if the first digit cannot be 0 or 9 and if the last digit is greater than or equal to 2 and less than or equal to 3? Repeated digits are allowed.

To determine

To find: How many seven digits can be formed?

Answer to Problem 17RE

1600000

Explanation of Solution

Given:

Using the digits 0, 1, 2, … 9, 7-digit numbers has to be formed if the first digit cannot be 0 or 9 and if the last digit is greater than or equal to 2 and less than or equal to 3? Repeated digits are allowed.

Calculation:

The generalized basic principle of counting.

we need to choose 7 digit number using 0, 1, 2 …9.

First digit cannot be 0 or 9.

Hence number of choices for 1^st digit $= 8$.

Number of choices for each 2^nd, 3,^rd, 4^th, 5^th and 6^th digit is 10.

Last digit is greater than or equal to 2 and less than or equal to 3.

Hence last digit can either be 2 or 3 only.

The number of options for last digit is 2.

Therefore the number of options for 7 digits is $8*10*10*10*10*10*2 = 1600000$.