From a standard deck of 52 distinct playing cards, how many different 5-card hands can be drawn if the order in which the cards are dealt does not matter?

Introduction / Context:
This is a classic card combinatorics problem that appears very frequently in probability and aptitude topics. When dealing with poker hands or any 5-card hands, we consider only which cards are in the hand, not the order in which they were drawn. Understanding how to count such hands using combinations is fundamental for more advanced probability questions involving cards, suits, and ranks.

Given Data / Assumptions:

• A standard deck has 52 distinct cards.
• We are drawing a hand of 5 cards.
• The order of cards in the hand does not matter.
• No card is repeated since a standard deck does not allow duplicates.
• Every 5-card subset is treated as one unique hand.

Concept / Approach:
Because order does not matter, we again use combinations. We must compute the number of ways to choose 5 cards out of 52. This is written as 52C5. The formula is:
nCr = n! / (r! * (n - r)!) So for this problem,
52C5 = 52! / (5! * 47!).

Step-by-Step Solution:
Step 1: Identify n = 52 and r = 5. Step 2: Apply the combination formula: 52C5 = 52! / (5! * 47!). Step 3: Write 52! / 47! as 52 * 51 * 50 * 49 * 48. Step 4: Compute numerator product: 52 * 51 * 50 * 49 * 48. Step 5: Divide by 5! = 120, simplifying step by step to reduce arithmetic mistakes. Step 6: After careful simplification, the result is 2598960. Step 7: Therefore, there are 2598960 distinct 5-card hands.

Verification / Alternative check:
This value 2598960 is a standard result in probability theory and card combinatorics. It is widely quoted in textbooks and online resources for the number of 5-card hands from a deck of 52 cards. You can also verify the value using a scientific calculator with nCr functionality. Any deviation from 2598960 indicates an arithmetic or formula error.

Why Other Options Are Wrong:
2589860: Close in magnitude but not equal to 52C5. 2598970: Only slightly different in the last digits, indicating a small calculation error. 2430803: Significantly smaller and not linked to the correct combination computation. Only 2598960 exactly matches the mathematically correct value for 52C5.

Common Pitfalls:
Common mistakes include treating each arrangement as different by using permutations instead of combinations, which would dramatically overcount the number of hands. Another mistake is performing the factorial division incorrectly, especially when handling large numbers. Breaking the calculation into manageable steps and cancelling terms before multiplying avoids overflow and reduces the chance of arithmetic error. Remember that for card hands, unless explicitly stated otherwise, order never matters, so combinations are the correct tool.

Final Answer:
The total number of distinct 5-card hands that can be drawn from a 52-card deck is 2598960.