Problem 1 · AMC 8 Stretch
Core
Counting & Probability
pigeonholecasework
You pick 5 cards from a big pile of cards that are each either red or blue. Show that no matter which 5 you grab, you are guaranteed to have at least 3 cards of the same color. (How many of one color are you guaranteed?)
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Answer: at least 3 of one color
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Hint 1 of 3
There are only two colors. Imagine two boxes: a red box and a blue box. Drop each card into the box for its color.
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Hint 2 of 3
You have 5 cards going into only 2 boxes. Could both boxes have 2 or fewer cards?
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Hint 3 of 3
If each box had at most 2 cards, you'd have at most \(2+2 = 4\) cards. But you have 5! So some box must have 3 or more.
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Approach: Pigeonhole — 5 cards into 2 color-boxes
- Make two boxes (the 'holes'): one for red cards, one for blue cards. Each of your 5 cards goes into the box matching its color.
- Could you avoid having 3 of one color? That would mean each box holds at most 2 cards. But \(2+2 = 4\), and you have 5 cards.
- So at least one box must hold 3 or more cards — giving 3 cards of the same color. You are guaranteed at least \(3\) of one color.
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