How many different five-card hands can be dealt from a standard deck of fifty-two cards?

To find the number of different five-card hands possible from a standard deck of fifty-two cards, you can use combinations. Combinations are used when the order of selection does not matter, which is the case with dealing cards since any order of the same cards is still the same hand.

The formula for combinations is C(n, k) = n! / [k!(n - k)!], where 'n' is the total number of items to choose from, 'k' is the number of items to choose, and '!' is the factorial of a number.

Here: - n = 52 (since there are 52 cards in a deck), - k = 5 (because we want to select 5 cards).

Therefore, the number of different five-card hands is:

C(52, 5) = 52! / [5!(52 - 5)!] = 52! / [5!47!] = (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1)

When you do the math:

52 × 51 × 50 × 49 × 48 = 311,875,200 5 × 4 × 3 × 2 × 1 = 120

So, dividing the two numbers:

311,875,200 / 120 = 2,598,960

Hence, there are 2,598,960 different five-card hands possible from a standard deck of fifty-two cards.