# PWN the SAT Q&A

Anonymous

## Anonymous asked:

Hi! How many positive integers less than 1000 do NOT have 7 as any digit? Is there a universal way to solve this? Thanks!

I don’t know what you mean by universal, but there’s definitely a shortcut!

Set up 3 blanks for each of the 3 digit places:

____  ____  ____

To fill in those blanks, we’re going to determine the number of choices we have for digits.

The hundreds place could be 1, 2, 3, 4, 5, 6, 8, 9, or 0. Anything but 7. That’s 9 choices.

__9__  _____  _____

The tens place is the same. It could be any digit but 7. And the units (ones) place is ALSO the same.

__9__  __9__  __9__

Now we multiply our choices together: 9 × 9 × 9 = 729.

At this point, you’re almost done. One of the possibilities we accounted for was the possibility of all 3 digits being 0. If that happens, it’s not a POSITIVE integer, so we can’t count it in our final tally.

729 ways you can place 3 digits without using a “7” - 1 for “000” = 728 positive integers less than 1000 that do NOT have 7 as any digit.

[For some more fun with counting, click here and here and here]

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