<p>This was in the PSAT blue book, and the only one I got wrong in math, and I still can't figure out how to get the right answer. The question is: How many positive integers less than 1000 do not have a 7 for any of the digits?</p>
<p>Disregard.</p>
<p>That’s not an option…
choices are
a)700
b)728
c)736
d)770
e)819</p>
<p>Mhm. Alrighty then. We have only three digit integers less than 1000. So that means we have three spots in the number.</p>
<p>For the first number, we can choose: 1, 2, 3, 4, 5, 6, not 7, 8, 9. That is 8 numbers.
For the second number, we can choose: 0, 1, 2, 3, 4, 5, 6, not 7 again, 8, 9. That’s 9 numbers.
For the third number, once again we can choose: 0, 1, 2…9. That adds up to 8, once more.</p>
<p>That’s for 3 digit integers.</p>
<p>Let’s move on to two digit integers.</p>
<p>First number has 8 options, second number has 9 options. 8 x 9 = 72.</p>
<p>And finally, an additional 8 numbers from 0 - 10, not including 7 or 0 (since 0 isn’t positive), that can also be integers that do not have 7 for any of their digits.</p>
<p>8 + 72 + 648 = 728</p>
<p>The answer is (B).</p>