Question

How many 7-digits phone numbers can be formed in the first digit cannot be 0 and any digit can be repeated?

Answer 1

Hint: We first describe the conditions where first digit cannot be 0. We find out the number of options we have for first digit. Then we find the number of options we have for all the other six digits as there is no restriction in repetition. We multiply to get the solution.

Complete step-by-step answer:
We need to find the number of 7-digits phone numbers that can be formed.
The only condition being the first digit can’t be 0.
As there is no other condition, the main criteria for getting those phone numbers is to fill up those 7 spots with numbers ranging from 0 to 9.
For the first digit’s place we can place only 1-9 as 0 is not allowed to place there.
So, there are 9 options for the first digit’s place.
Now we have the permission to use one digit as many times as we want.
For all the other six places of the phone number, we have no restrictions.
So, we can place 1-9 as 0 is allowed to place in all the other digits.
Therefore, for all the other six places we have 10 options.
All the process of filling up places is independent of each other. That’s why we take multiplication of those outcome numbers.
The final result is $9\times {{10}^{6}}$. This is equal to 9000000.
Therefore, there are a total 9000000 number of ways we can get 7-digits phone numbers.

Note: We can also find all the ways without any restriction which is ${{10}^{7}}$. Now we find all the ways we can have numbers starting with 0 which is ${{10}^{6}}$. Then we get the answer by subtraction which is ${{10}^{7}}-{{10}^{6}}=9\times {{10}^{6}}$.