From a group of 8 people, how many ways can a committee of 4 be formed?

From a group of 8 people, how many ways can a committee of 4 be formed?

Correct Answer: 70

Step 1: Understand that we need to choose 4 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, C, D is the same as choosing D, C, B, 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 (4).
Step 4: Plug in the values into the formula: C(8, 4) = 8! / (4! * (8 - 4)!).
Step 5: Calculate 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1, 4! = 4 × 3 × 2 × 1, and (8 - 4)! = 4! = 4 × 3 × 2 × 1.
Step 6: Simplify the calculation: C(8, 4) = (8 × 7 × 6 × 5) / (4 × 3 × 2 × 1).
Step 7: Calculate the numerator: 8 × 7 × 6 × 5 = 1680.
Step 8: Calculate the denominator: 4 × 3 × 2 × 1 = 24.
Step 9: Divide the numerator by the denominator: 1680 / 24 = 70.
Step 10: Conclude that there are 70 different ways to form a committee of 4 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.