From a group of 8 people, how many ways can a committee of 3 be formed?

From a group of 8 people, how many ways can a committee of 3 be formed?

Correct Answer: 56

Step 1: Understand that we need to choose 3 people from a group of 8.
Step 2: Recognize that the order in which we choose the people does not matter (i.e., choosing person A, B, and C is the same as choosing C, B, and A).
Step 3: Use the combination formula C(n, r) = n! / (r! * (n - r)!), where n is the total number of people (8) and r is the number of people to choose (3).
Step 4: Plug in the values into the formula: C(8, 3) = 8! / (3! * (8 - 3)!) = 8! / (3! * 5!).
Step 5: Calculate 8! = 8 × 7 × 6 × 5!, so we can simplify: C(8, 3) = (8 × 7 × 6) / (3 × 2 × 1).
Step 6: Calculate the numerator: 8 × 7 × 6 = 336.
Step 7: Calculate the denominator: 3 × 2 × 1 = 6.
Step 8: Divide the numerator by the denominator: 336 / 6 = 56.
Step 9: Conclude that there are 56 different ways to form a committee of 3 from 8 people.

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