Problem 18 · 2015 Math Kangaroo
Stretch
Counting & Probability
caseworkcomplementary-counting
3 green apples, 5 yellow apples, 7 green pears and 2 yellow pears are in a sack. Without looking, Sebastian takes either an apple or a pear out of the sack. How many pieces of fruit must he take out of the sack to be sure of having at least one apple and one pear of the same colour?
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Answer: E — 13
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Hint 1 of 2
Think of the unluckiest case: how many fruit could he pull out and still NOT have an apple and a pear of the same colour?
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Hint 2 of 2
He could keep drawing green pears and yellow apples forever without ever matching a colour across the two fruit types.
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Approach: find the largest unlucky collection with no same-colour apple+pear, then add one
- He wins when he holds an apple and a pear of the same colour: green-with-green or yellow-with-yellow.
- The biggest collection that still avoids this takes all 7 green pears and all 5 yellow apples — green pears with no green apple, yellow apples with no yellow pear — that's 12 fruit and still no match.
- Any 13th fruit must be a green apple or a yellow pear, which completes a same-colour pair, so he needs to take 13.
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