Question: From a standard 52-card deck, how many 2-card hands consist entirely of red cards? There are…

From a standard 52-card deck, how many 2-card hands consist entirely of red cards? There are possible hands.

Solution

To find how many 2-card hands consist entirely of red cards from a standard 52-card deck, we need to follow these steps: First, note that there are two suits of red cards in a standard deck: hearts and diamonds. Each suit has 13 cards. The total number of red cards is: \[ 13 + 13 = 26 \] Next, we need to determine how many 2-card combinations can be made from these 26 cards. We use the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items to choose from (26 red cards), and \( r \) is the number of items to choose (2 cards). \[ \binom{26}{2} = \frac{26!}{2!(26-2)!} = \frac{26 \times 25}{2 \times 1} \] Calculating the result: \[ = \frac{650}{2} = 325 \] So, there are 325 possible 2-card hands that consist entirely of red cards.