Everyone, this is problem three on the grade 11 and 12 2025 test. Vasily has 20 different colored balls, yellow, green, blue, and black. Of these, exactly 17 are not green, 15 are not black, and 12 are not yellow. How many balls are blue? We have A, 8, B, 7, C, 6, D, 4, E, 3. Try this yourself first. And now we may start. Okay, so the wording is kind of roundabout, but let's start with the information we do know. So we have 20 total balls, and we know that 17 of them are not green. And yeah, 17 are not green. We know that 15 are not black. We know that 12 are not yellow. And we'd have like an unknown amount of blues. So, yeah, I said this was roundabout, and it's because they're giving us information on what's not green. So if I want to determine like the actual number of green balls, well, what do I do? I can just, you know, figure it out by complementary counting. If I know the things that are like not in my set that I'm counting, and I sort of know the total set, then I can just subtract all the stuff that's in the not set and obtain the set that I'm looking for. So yeah, after all that jargon, we obtained that there are actually 20 minus 17 green balls, which is three greens. By like an identical line of logic, there are 20 minus 15 black balls. So that means there are five black balls. Then there are, you know, 20 minus 12 yellow balls. So eight yellows. And finally, we can just sort of count these up to determine the number of not blue balls. And that's 3 plus 5 plus 8 equal to 16 not blue balls. Which then indicates that there are four blue balls. And that's our answer choice. There are four blue balls.

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