How many ways can 5 different cards be chosen from a deck of 52 cards? (2019)

How many ways can 5 different cards be chosen from a deck of 52 cards? (2019)

Step 1: Understand that we want to choose 5 different cards from a total of 52 cards.
Step 2: Recognize that this is a combination problem because the order of the cards does not matter.
Step 3: Use the combination formula, which is written as nCr, where n is the total number of items (52 cards) and r is the number of items to choose (5 cards).
Step 4: The combination formula is nCr = n! / (r! * (n - r)!), where '!' denotes factorial, which means multiplying a number by all the positive integers below it.
Step 5: Plug in the values: 52C5 = 52! / (5! * (52 - 5)!) = 52! / (5! * 47!).
Step 6: Calculate 52! / (47!) which simplifies to 52 * 51 * 50 * 49 * 48 because the 47! cancels out.
Step 7: Calculate 5! which is 5 * 4 * 3 * 2 * 1 = 120.
Step 8: Now divide the result from Step 6 by the result from Step 7: (52 * 51 * 50 * 49 * 48) / 120.
Step 9: Perform the multiplication: 52 * 51 * 50 * 49 * 48 = 311875200.
Step 10: Finally, divide 311875200 by 120 to get 2598960.

Combinatorics – The problem tests the understanding of combinations, specifically how to calculate the number of ways to choose a subset of items from a larger set without regard to the order of selection.
Binomial Coefficient – The use of the binomial coefficient notation (nCr) to represent the number of combinations, which is crucial for solving problems involving selection.