From a group of 10 people, how many ways can a committee of 4 be formed?

From a group of 10 people, how many ways can a committee of 4 be formed?

Correct Answer: 210

Step 1: Understand that we need to choose 4 people from a group of 10.
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, which is written as nCr, where n is the total number of people and r is the number of people to choose. Here, n = 10 and r = 4.
Step 4: The combination formula is nCr = n! / (r! * (n - r)!), where '!' denotes factorial (the product of all positive integers up to that number).
Step 5: Calculate 10C4 using the formula: 10C4 = 10! / (4! * (10 - 4)!) = 10! / (4! * 6!).
Step 6: Simplify the factorials: 10! = 10 × 9 × 8 × 7 × 6!, so we can cancel 6! in the numerator and denominator.
Step 7: Now we have: 10C4 = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1).
Step 8: Calculate the numerator: 10 × 9 × 8 × 7 = 5040.
Step 9: Calculate the denominator: 4 × 3 × 2 × 1 = 24.
Step 10: Divide the numerator by the denominator: 5040 / 24 = 210.
Step 11: Conclude that there are 210 different ways to form a committee of 4 from a group of 10 people.

Combinatorics – The study of counting, arrangements, and combinations of objects.
Binomial Coefficient – The formula used to determine the number of ways to choose a subset of items from a larger set, denoted as nCr.