How many ways can 4 different fruits be selected from 10 available fruits?

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Question: How many ways can 4 different fruits be selected from 10 available fruits?

Options:

Correct Answer: 210

Solution:

The number of ways to select 4 fruits from 10 is C(10, 4) = 10! / (4! × (10-4)!) = 210.

How many ways can 4 different fruits be selected from 10 available fruits?

How many ways can 4 different fruits be selected from 10 available fruits?

Step 1: Understand that we want to choose 4 different fruits from a total of 10 fruits.
Step 2: Recognize that the order in which we select the fruits does not matter. This means we will use combinations, not permutations.
Step 3: The formula for combinations is C(n, r) = n! / (r! × (n - r)!), where n is the total number of items, and r is the number of items to choose.
Step 4: In our case, n = 10 (the total fruits) and r = 4 (the fruits we want to select).
Step 5: Plug the values into the formula: C(10, 4) = 10! / (4! × (10 - 4)!)
Step 6: Simplify the equation: C(10, 4) = 10! / (4! × 6!)
Step 7: Calculate 10! (which is 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) but notice that 6! will cancel out in the denominator.
Step 8: So we only need to calculate: (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1).
Step 9: Calculate the numerator: 10 × 9 = 90, then 90 × 8 = 720, then 720 × 7 = 5040.
Step 10: Calculate the denominator: 4 × 3 = 12, then 12 × 2 = 24, then 24 × 1 = 24.
Step 11: Now divide the numerator by the denominator: 5040 / 24 = 210.
Step 12: Therefore, the number of ways to select 4 different fruits from 10 available fruits is 210.

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.