Round #1064

How many distinct 5-card hands can be dealt from a standard 52-card deck? Give the exact integer.

## Step 1: Understand the problem To find the number of distinct 5-card hands that can be dealt from a standard 52-card deck, we need to calculate the combinations of 52 cards taken 5 at a time. ## Step 2: Calculate the combination The formula for combinations is given by C(n, k) = n! / [k!(n-k)!], where n is the total number of items, k is the number of items to choose, and "!" denotes factorial, which is the product of all positive integers up to that number. ## Step 3: Apply the formula For our problem, n = 52 (total number of cards) and k = 5 (number of cards in a hand). So, we calculate C(52, 5) = 52! / [5!(52-5)!]. ## Step 4: Simplify the expression C(52, 5) = 52! / [5! * 47!] = (52 * 51 * 50 * 49 * 48) / (5 * 4 * 3 * 2 * 1), because the factorials in the numerator and the denominator cancel out all but the first 5 numbers in the numerator. ## Step 5: Perform the calculation Calculate the numerator: 52 * 51 * 50 * 49 * 48 = 311,875,200. Then calculate the denominator: 5 * 4 * 3 * 2 * 1 = 120. Finally, divide the numerator by the denominator: 311,875,200 / 120 = 2,598,960. The final answer is: $\boxed{2598960}$

52 choose 5, or 2,598,960.