How many ways can a committee of 3 be formed from 10 people?

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Question: How many ways can a committee of 3 be formed from 10 people?

Options:

Correct Answer: 120

Solution:

The number of ways to choose 3 people from 10 is C(10, 3) = 10! / (3!(10-3)!) = 120.

How many ways can a committee of 3 be formed from 10 people?

How many ways can a committee of 3 be formed from 10 people?

Step 1: Understand that we need to choose 3 people from a group of 10 people.
Step 2: Recognize that this is a combination problem, where 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 people (10) and r is the number of people to choose (3).
Step 4: Plug in the values into the formula: C(10, 3) = 10! / (3!(10-3)!).
Step 5: Simplify the formula: C(10, 3) = 10! / (3! * 7!).
Step 6: Calculate 10! = 10 × 9 × 8 × 7! (the 7! cancels out).
Step 7: Now we have C(10, 3) = (10 × 9 × 8) / (3 × 2 × 1).
Step 8: Calculate the numerator: 10 × 9 × 8 = 720.
Step 9: Calculate the denominator: 3 × 2 × 1 = 6.
Step 10: Divide the numerator by the denominator: 720 / 6 = 120.
Step 11: Conclude that there are 120 different ways to form a committee of 3 from 10 people.

Combinatorics – The study of counting, arrangements, and combinations of objects.
Binomial Coefficient – The formula C(n, k) = n! / (k!(n-k)!) used to determine the number of ways to choose k elements from a set of n elements.