How many 4 card hands are possible with a 13-card deck?

To find out how many 4-card hands are possible from a 13-card deck, we use the combination formula, which is highlighted as follows: the number of ways to choose \( r \) items from \( n \) items without regard to the order is given by \( C(n, r) = \frac{n!}{r!(n-r)!} \). Using this formula, we plug in \( n = 13 \) and \( r = 4 \): \[ C(13, 4) = \frac{13!}{4!(13-4)!} = \frac{13!}{4! \cdot 9!} \] This simplifies to: \[ C(13, 4) = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = 715 \] So, there are 715 different 4-card hands possible with a 13-card deck. Did you know that in many card games, the strategy can change drastically based on the number of possible hands? Understanding the combinations can give you an edge, whether you're bluffing in poker or making strategic plays in bridge. The more you know about moves and counts, the better you'll play! For those looking to deepen their understanding of combinatorial mathematics beyond just card games, books like "Discrete Mathematics and Its Applications" by Rosen provide a wealth of knowledge. It offers a great mix of theory and practical examples, making complex concepts easier to digest. Perfect for math enthusiasts and gamers alike!