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

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: The formula for combinations is nCr = n! / (r! * (n - r)!), where '!' denotes factorial, which is the product of all positive integers up to that number.
Step 5: In our case, n = 10 and r = 4. So we need to calculate 10C4.
Step 6: Plug the values into the formula: 10C4 = 10! / (4! * (10 - 4)!) = 10! / (4! * 6!).
Step 7: Calculate the factorials: 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1, 4! = 4 × 3 × 2 × 1, and 6! = 6 × 5 × 4 × 3 × 2 × 1.
Step 8: Simplify the expression: 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.

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