If a committee of 4 is to be formed from 8 people, how many different committees

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Question: If a committee of 4 is to be formed from 8 people, how many different committees can be formed?

Options:

Correct Answer: 70

Solution:

The number of ways to form a committee of 4 from 8 is C(8, 4) = 8! / (4! × (8-4)!) = 70.

If a committee of 4 is to be formed from 8 people, how many different committees can be formed?

If a committee of 4 is to be formed from 8 people, how many different committees can be formed?

Step 1: Understand that we need to choose 4 people from a group of 8 people.
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 - 4) which equals 4, so we have C(8, 4) = 8! / (4! × 4!).
Step 6: Calculate 8! (which is 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) and 4! (which is 4 × 3 × 2 × 1).
Step 7: Substitute the factorial values into the equation: C(8, 4) = 40320 / (24 × 24).
Step 8: Calculate 24 × 24 which equals 576.
Step 9: Divide 40320 by 576 to get the final answer: 40320 / 576 = 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 – A way to calculate the number of ways to choose a subset of items from a larger set, denoted as C(n, k) or nCk.