Question

How many positive integers less than 1000 do not have 7 as any digit? I know the answer, but I don't know how to show the work and solve it out.

(A) 700
(B) 728
(C) 736
(D) 770
(E) 819

Answer 1

Find the number of positive integers that do have 7 as any digit first. There is one one-digit number, there are 18 two-digit numbers, and there are 244 three-digit numbers. So, there are a total of 263 numbers less than 1000 with a 7 as a digit. There are a total of 999 numbers less than 1000, so 999-263=736 numbers less than 1000 without 7 as a digit.

Answer 2

Well it says that the correct answer is B, 728

Answer 3

I might have missed some :P

Answer 4

Well okays and thank you for helping!

Answer 5

Oh! I figured it out. I wasn't counting some three-digit numbers, there are 252 of them. So it is 728.

Answer 6

Number of three-digit numbers with a seven = 19*8+100, I had done 18*8+100, forgetting about numbers of the form n07 (107, 207, etc)

Answer 7

Okay thank you! So did you just basically count all those numbers, or is there a faster way to solve this? Because this is from a practice PSAT test and it's timed

Answer 8

You just think about patterns. For three digit numbers, for each hundred, you're going to have the 18 two digit numbers and 1 one digit numbers that you already counted, except for the 700s they will all count, so that's how you get 19*8+100. For the two digit numbers it's just 8+10, 8 that are like 17, 27, etc, and ten that are the 70s.

Answer 9

Oh I see, thank you!