How many numbers are there between 99 and 1000 having at least one of their digits 7?

Total number of 3-digit numbers having at least one of their digits as 7 = (Total numbers of three-digit numbers) – (Total number of 3-digit numbers in which 7 does not appear at all) ….. (1)

Total numbers of three-digit numbers:

Between 99 and 1000 all the numbers are 3-digit numbers.

Also, we have numbers from 0 to 9.

So total three digits numbers can be found as:

1^st place can be filled in 10 ways.

2^nd place can be filled in 10 ways.

3^rd place can be filled in 9 ways, if 0 will appear on 3^rd place the number will be a two-digit number.

Total ways = 9× 10× 10

= 900 ways

Total number of 3-digit numbers in which 7 does not appear at all:

Now we are not considering 7.

So total three digits numbers can be found as:

1^st place can be filled in 9 ways.

2^nd place can be filled in 9 ways.

3^rd place can be filled in 8 ways, if 0 will appear on 3^rd place the number will be a two-digit number.

Total ways = 8× 9× 9

= 648 ways

From (1),

Total number of 3-digit numbers having at least one of their digits as 7 = 900 – 648

= 252