How many ways can you choose 3 items from a set of 10?

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Question: How many ways can you choose 3 items from a set of 10?

Options:

Correct Answer: 120

Solution:

The number of combinations is calculated as 10! / (3!(10-3)!) = 120.

How many ways can you choose 3 items from a set of 10?

How many ways can you choose 3 items from a set of 10?

Step 1: Understand that we want to choose 3 items from a total of 10 items.
Step 2: Recognize that the order in which we choose the items does not matter, so we will use combinations.
Step 3: The formula for combinations is given by: 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 (total items) and r = 3 (items to choose).
Step 5: Plug the values into the formula: C(10, 3) = 10! / (3!(10 - 3)!).
Step 6: Simplify the formula: C(10, 3) = 10! / (3! * 7!).
Step 7: Calculate 10! (which is 10 x 9 x 8 x 7!) and notice that 7! cancels out in the numerator and denominator.
Step 8: Now we have: C(10, 3) = (10 x 9 x 8) / (3 x 2 x 1).
Step 9: Calculate the numerator: 10 x 9 x 8 = 720.
Step 10: Calculate the denominator: 3 x 2 x 1 = 6.
Step 11: Divide the numerator by the denominator: 720 / 6 = 120.
Step 12: Therefore, there are 120 different ways to choose 3 items from a set of 10.

Combinations – The concept of combinations involves selecting items from a larger set where the order of selection does not matter.
Factorial – Understanding factorial notation (n!) is crucial for calculating combinations, as it represents the product of all positive integers up to n.