In how many ways can 3 different fruits be selected from a basket of 7 fruits? (

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Question: In how many ways can 3 different fruits be selected from a basket of 7 fruits? (2014)

Options:

Correct Answer: 35

Exam Year: 2014

Solution:

The number of ways to select 3 from 7 is C(7, 3) = 35.

In how many ways can 3 different fruits be selected from a basket of 7 fruits? (2014)

In how many ways can 3 different fruits be selected from a basket of 7 fruits? (2014)

Step 1: Understand that we need to choose 3 different fruits from a total of 7 fruits.
Step 2: Recognize that this is a combination problem because the order of selection does not matter.
Step 3: Use the combination formula C(n, r) = n! / (r! * (n - r)!), where n is the total number of items (7 fruits) and r is the number of items to choose (3 fruits).
Step 4: Plug in the values into the formula: C(7, 3) = 7! / (3! * (7 - 3)!).
Step 5: Calculate 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1, which equals 5040.
Step 6: Calculate 3! = 3 × 2 × 1, which equals 6.
Step 7: Calculate (7 - 3)! = 4! = 4 × 3 × 2 × 1, which equals 24.
Step 8: Substitute these values back into the formula: C(7, 3) = 5040 / (6 * 24).
Step 9: Calculate 6 * 24 = 144.
Step 10: Finally, divide 5040 by 144 to get C(7, 3) = 35.

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.