Grades 9-10 Video Solutions 2025-2025_9-10

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Problem 15. Sarah has a bag of 18 balls numbered from 1 to 18. What's the smallest number of balls that Sarah should remove in order to guarantee that  she's removed at least 3 balls with prime numbers on them? So first let's list out the primes less than 18. So there are 7 and these are 2, 3, 5, 7, 11, 13,  and 17. Now we need to guarantee that Sarah has removed at least 3 balls. So this means we can't just think about any case that would allow her to remove  3 balls of prime numbers. We have to make sure that she will for sure remove 3 balls of prime numbers even in the worst case. So what would the worst case  be? The worst case would be when before we get our 3 balls with prime numbers we also get a lot of things that are not useful. So the 11 balls that  don't have prime numbers on them are the balls that are not useful for us. And the worst case would be having to draw out all of those before we get our 3  primes. But after we've drawn out all of those then everything left is a prime. So just getting 3 more will ensure that we get our 3 primes. So we have 18 minus 7  or 11 non-prime integers and we would have to take out all of those and then 3 prime integers and that makes 14 balls in total that need to be taken out  in the worst case. So if we take out 14 that guarantees that in any case we will remove at least 3 balls of prime numbers.

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