In how many ways can 4 different fruits be selected from a basket of 10 fruits?

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Question: In how many ways can 4 different fruits be selected from a basket of 10 fruits?

Options:

Correct Answer: 210

Solution:

The number of ways to choose 4 fruits from 10 is given by 10C4 = 210.

In how many ways can 4 different fruits be selected from a basket of 10 fruits?

In how many ways can 4 different fruits be selected from a basket of 10 fruits?

Step 1: Understand that we need to choose 4 different fruits from a total of 10 fruits.
Step 2: Recognize that this is a combination problem because the order of selection does not matter.
Step 3: 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.
Step 4: In this case, n = 10 (total fruits) and r = 4 (fruits to choose).
Step 5: The combination formula is nCr = n! / (r! * (n - r)!), where '!' denotes factorial, which is the product of all positive integers up to that number.
Step 6: Calculate 10C4 using the formula: 10C4 = 10! / (4! * (10 - 4)!) = 10! / (4! * 6!).
Step 7: Simplify the factorials: 10! = 10 × 9 × 8 × 7 × 6!, so we can cancel 6! in the numerator and denominator.
Step 8: Now we have: 10C4 = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1).
Step 9: Calculate the numerator: 10 × 9 × 8 × 7 = 5040.
Step 10: Calculate the denominator: 4 × 3 × 2 × 1 = 24.
Step 11: Divide the numerator by the denominator: 5040 / 24 = 210.
Step 12: Conclude that there are 210 different ways to select 4 fruits from 10.

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.