Question

How many \[5\] digit phone numbers can be formed with the digits $ 0 $ , $ 1 $ , $ 2 $ …. $ 9 $ , if each number starts with $ 35 $ and no digit appears more than once?

Answer 1

Hint: In the given question we are required to find out the number of combinations of $ 5 $ digit phone numbers when we are given the additional information or conditions regarding the same. Each number must start with $ 35 $ and no digit should appear more than once in the phone number. We have the digits $ 0 $ , $ 1 $ , $ 2 $ .... $ 9 $ so as to form a phone number.

Complete step-by-step answer:
So, we are to create \[5\] digit phone numbers. So, we have $ 10 $ digits with us to fill up the $ 5 $ positions. We are given the condition that each number starts with $ 35 $ . Also, no digit should appear more than once in any phone number. So, we can say that there will be no $ 3 $ or $ 5 $ in the remaining three places of the phone number. Hence, we have to fill up the remaining three places in the phone number by the digits $ 0 $ , $ 1 $ , $ 2 $ , $ 4 $ , $ 6 $ , $ 7 $ , $ 8 $ and $ 9 $ .
So, there are a total of $ 8 $ digit options.
Now, Number of options to fill up one of the three places in the phone number is $ 8 $ .
Then, the digit already used up cannot be used again as no repetition of digit is allowed.
Hence, the number of options to fill up one of the remaining two places is $ 7 $ .
Also, the number of options to fill up the last place in the phone number is $ 6 $ .
So, using the fundamental principle of counting, we get,
The total number of \[5\] digit phone numbers can be formed with the given conditions are $ 8 \times 7 \times 6 $ .
 $ \Rightarrow 336 $
So, the total number of \[5\] digit phone numbers can be formed with the digits $ 0 $ , $ 1 $ , $ 2 $ …. $ 9 $ , if each number starts with $ 35 $ and no digit appears more than once is $ 336 $ .

Note: In such a problem, one must know the multiplication rule of counting as consequent events are occurring. One must take care while doing the calculations and should recheck them so as to be sure of the final answer.