Question

How many 6-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?

Answer 1

Hint: For 3rd, 4th ,5th and 6th place, We have to choose digits from 0 to 9 but the digit 6 and 7 cannot be taken because repetition is not allowed. So, we have 8 possible digits for 3rd place. As repetition is not allowed, so we have 7 digits left for 4th place. Similarly, we have 6 digits and 5 digits left for 5th place and 6th place. The total number of 6 digit telephone numbers \[=8\times 7\times 6\times 5=56\times 30=1680\] .

Complete step-by-step answer:
Here, we have to form a 6-digit number and we have to choose any number from 0 to 9 for all the 6 digits. But it is given that the number should start with 67, so the first two places are fixed with the digit 6 and 7 respectively. Now, we have to find the number of digits for the 3rd place, 4th place, 5th place, and 6th place as shown in the figure below.

As the digit 6 and 7 is already taken and repetition is not allowed. So, for 3rd place, we have 8 digits possible. Now we have chosen one digit for the 3rd place so that digit should not be chosen for the 4th place because repetition is not allowed. So, for the 4th place, we have 7 digits possible.
Similarly, for the 5th place and 6th place we have 6 digits and 5 digits possible.
The total number of 6 digit telephone numbers \[=8\times 7\times 6\times 5=56\times 30=1680\] .
Hence, the total number of 6 digit numbers is 1680.

Note: In this question, one can include the digit 6 and 7 for the3rd, 4th ,5th and 6th place. If we do so, then repetition will be done as this number is starting with 67. But according to the question, we have that there should not be any repetition. So, we have to exclude the digit 6 and 7.