Using the pigeonhole principle, how many items are needed to guarantee at least one pair of identical colors if there are 5 colors?

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Prepare for the AMSOC 26-003 Module A Test. Utilize flashcards and multiple choice questions with hints and explanations. Ace your exam!

Multiple Choice

Using the pigeonhole principle, how many items are needed to guarantee at least one pair of identical colors if there are 5 colors?

Explanation:
The pigeonhole principle tells us that if there are more items than colors (pigeonholes), at least one color must be used twice. With five colors, you could place five items so that each uses a different color, avoiding any pair. But add one more item, and there are only five colors to choose from, so two items must share a color. That guarantees a pair. Since five items can be colored without any duplicates, six is the smallest number that ensures a pair.

The pigeonhole principle tells us that if there are more items than colors (pigeonholes), at least one color must be used twice. With five colors, you could place five items so that each uses a different color, avoiding any pair. But add one more item, and there are only five colors to choose from, so two items must share a color. That guarantees a pair. Since five items can be colored without any duplicates, six is the smallest number that ensures a pair.

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