Question

How many seven-digit phone numbers can be formed if the first digit cannot be 0 and repetition of digits is not permitted.

Answer 1

In this problem, we have to find the number of ways of seven-digit phone numbers can be formed if the first digit cannot be 0 and repetition of digits is not permitted. We can first take the number from 0 to 9. We know that the first digit should not be zero, so we will have 9 ways, i.e. 1-9, then the second digit will include the 0, and we have remaining 8 from the first digit, therefore, we will have 9 way, similarly we can continue and multiply the number of ways to get the total number of ways.

Complete step by step solution:
We know that we have to find the number of ways seven-digit phone numbers can be formed if the first digit cannot be 0 and repetition of digits is not permitted.
We have numbers from 0 to 9.
We can now find the number of ways for the first digit.
We can see that we have 9 ways for the first digit excluding 0, i.e. 1 – 9.
We can now find the number of ways for the second digit.
We know that we have 8 remaining numbers from the first digit, but here we can include 0, so we will have 9 ways, i.e. 0, remaining 8 numbers from the first term.
We can now find the third term possibilities, as we have 0 plus remaining 7 numbers from the second digit. i.e. 8 ways
We know that the fourth term possibility will be 0 plus the remaining 6 numbers from the third digit. i.e. 7 ways.
We know that the fifth term possibility will be 0 plus the remaining 5 numbers from the fourth digit. i.e. 6 ways.
We know that the sixth term possibility will be 0 plus the remaining 4 numbers from the fifth digit. i.e. 5 ways.
We know that the seventh term possibility will be 0 plus the remaining 5 numbers from the sixth digit. i.e. 4 ways.
We can now multiply the number of ways for each term, we get
$\Rightarrow 9\times 9\times 8\times 7\times 6\times 5\times 4=544320$ways.
Therefore, we have 544320 ways of seven-digit phone numbers can be formed if the first digit cannot be 0 and repetition of digits is not permitted.

Note: Students make mistakes like including the zero in the first digit, as we are given, we should not include it. So, we have to exclude it and take the number from 1 to 9 and the second digit can include the zero and the remaining terms from the first term, as similarly every terms are found