How many ways can you choose 3 fruits from a basket of 5 different fruits?

How many ways can you choose 3 fruits from a basket of 5 different fruits?

Step 1: Understand that you have 5 different fruits in a basket.
Step 2: You want to choose 3 fruits from these 5 fruits.
Step 3: Recognize that the order in which you choose the fruits does not matter (choosing apple, banana, cherry is the same as choosing cherry, banana, apple).
Step 4: Use the combination formula, which is written as nCr, where n is the total number of items (fruits) and r is the number of items to choose. Here, n = 5 and r = 3.
Step 5: The combination formula is nCr = n! / (r! * (n - r)!), where '!' means factorial (the product of all positive integers up to that number).
Step 6: Calculate 5C3 using the formula: 5C3 = 5! / (3! * (5 - 3)!)
Step 7: Calculate the factorials: 5! = 5 × 4 × 3 × 2 × 1 = 120, 3! = 3 × 2 × 1 = 6, and 2! = 2 × 1 = 2.
Step 8: Substitute the factorials into the formula: 5C3 = 120 / (6 * 2).
Step 9: Calculate the denominator: 6 * 2 = 12.
Step 10: Now divide: 120 / 12 = 10.
Step 11: Conclude that there are 10 different ways to choose 3 fruits from the basket of 5.

Combinatorics – The question 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.