Question

Tim wrote seven – digits phone numbers on a piece of paper. He later tore the paper accidentally and the last two digits were lost. What is the maximum number of arrangements of two digits, using the digits 0 through 9, that he could use in the attempt to reconstruct the correct phone number ?

Answer 1

Hint: From the question itself we can understand that we have to use permutations and combinations. 0 through 9 is 10 digits. The last two digits can be anything. The last two digits can be the same which means a number may be repeated. Or those two can be different which means they are two different digits. Now let us apply this logic.

Complete step by step answer:
There are seven digits. Out of which, the last two are lost.
So we should focus on the last two digits. The ${{5}^{th}}$ digit can be any number from 0 , 1 , 2 , 3 , 4 , 5 , 6 ,7 , 8 ,9. So it has 10 total possibilities. So I can arrange it in 10 different ways.
Now the ${{6}^{th}}$ digit can also be any number from 0 , 1 , 2 , 3 , 4 , 5 , 6 ,7 , 8 , 9. In a phone – number there are no restrictions that a number can not be repeated. So even the ${{6}^{th}}$ digit has 10 different possibilities.
So the total number of ways is $10\times 10=100$ ways.
$\therefore $ Hence , the maximum number of arrangements of two digits, using the digits 0 through 9, that he could use in the attempt to reconstruct the correct phone number is 100 .

Note:
We have to be very careful when solving questions from permutations and combinations. There is a lot of logic in each question. And the question has to be read and understood clearly. The logic can only be achieved through practice. We should not memorize any kind of formula in permutations and combinations. Logic must be understood and then applied.