# How many numbers are there between 100 and 1000 in which all the digits are distinct?

How many numbers are there between 100 and 1000 in which all the digits are distinct?

My mathematics textbook says the answer is 648.

My analysis:-

The number at the hundred's place can be chosen in 9 ways(as zero is not possible)

The number at the ten's place can be chosen in 9 ways(the number at the hundred's place is no longer available but zero is now available).

The number at the one's place can be chosen in 8 ways(the number at the hundred's and ten's place is not available but zero is available).

So by the multiplication rule, the total number of ways would seem to be 9*9*8=648.

But the hundred's place can have 1 at its place, ten's can have 0 and ones can have 0. So 100 is a possibility but the question is asking for numbers between 100 and 1000.As a result of that, 100 has to be removed. Therefore, the correct answer is 648-1=647.

The answer $648$ is correct.