At a card party, someone brings out a deck of bridge cards (4 suits with 13 cards in each). N people each pick 2 cards from the deck and hold onto them. What is the minimum value of N that guarantees at least 2 people have the same combination of suits?
Question
At a card party, someone brings out a deck of bridge cards (4 suits with 13 cards in each). N people each pick 2 cards from the deck and hold onto them. What is the minimum value of N that guarantees at least 2 people have the same combination of suits?
Solution
To solve this problem, we need to understand the concept of the Pigeonhole Principle. This principle states that if you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon. In this case, the 'pigeons' are the people and the 'pigeonholes' are the possible combinations of suits.
Step 1: Calculate the total number of possible combinations of suits. Since there are 4 suits, and each person picks 2 cards, the total number of combinations is 4 choose 2, which equals 6 (these combinations are: Spades-Hearts, Spades-Diamonds, Spades-Clubs, Hearts-Diamonds, Hearts-Clubs, Diamonds-Clubs).
Step 2: Apply the Pigeonhole Principle. To guarantee that at least 2 people have the same combination of suits, we need to have more 'pigeons' (people) than 'pigeonholes' (combinations of suits). Therefore, the minimum value of N (people) that guarantees at least 2 people have the same combination of suits is 7.
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