You can form exactly 126 committees of 4 members from a group of 9 people.
Understanding Combinations
When forming a committee, the order in which members are selected does not matter. For example, selecting person A, then B, then C, then D results in the same committee as selecting B, then A, then D, then C. This type of selection, where the order of items chosen is irrelevant, is known as a combination.
Combinations differ significantly from permutations, where the order does matter (e.g., arranging books on a shelf or determining race finishes).
| Feature | Combinations | Permutations |
|---|---|---|
| Order | Does NOT matter | DOES matter |
| Example | Selecting a committee, choosing lottery numbers | Arranging letters, ordering a queue |
| Formula | C(n, k) = n! / (k! * (n-k)!) | P(n, k) = n! / (n-k)! |
The Combination Formula
To calculate the number of combinations, we use the following formula:
*C(n, k) = n! / (k! (n-k)!)**
Where:
- C(n, k) represents the number of combinations.
- n is the total number of items to choose from (in this case, 9 people).
- k is the number of items to choose (in this case, 4 committee members).
- ! denotes the factorial, meaning the product of all positive integers less than or equal to that number (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120).
Step-by-Step Calculation
Let's apply the combination formula to determine how many committees of 4 members can be formed from 9 people.
-
Identify n and k:
- Total number of people (n) = 9
- Number of members for the committee (k) = 4
-
Substitute values into the formula:
C(9, 4) = 9! / (4! (9-4)!)
C(9, 4) = 9! / (4! 5!) -
Expand the factorials:
C(9, 4) = (9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / ((4 × 3 × 2 × 1) × (5 × 4 × 3 × 2 × 1)) -
Simplify the expression:
Notice that 5! appears in both the numerator and the denominator, allowing us to cancel it out:
C(9, 4) = (9 × 8 × 7 × 6) / (4 × 3 × 2 × 1) -
Perform the multiplication and division:
Numerator: 9 × 8 × 7 × 6 = 3024
Denominator: 4 × 3 × 2 × 1 = 24
C(9, 4) = 3024 / 24
C(9, 4) = 126
Thus, applying the combination formula to select 4 members from 9 people confirms that a committee of 4 members can be formed from 9 people in precisely 126 distinct ways.
Practical Applications of Combinations
The concept of combinations extends far beyond forming committees and is widely used in various fields:
- Probability: Calculating the odds in card games like poker or drawing lottery numbers. For instance, determining the number of possible 5-card hands from a deck of 52 cards.
- Statistics: In sampling, to determine the number of ways to select a sample from a larger population.
- Computer Science: In algorithm design and data structure optimization.
- Quality Control: Selecting a subset of items for inspection from a larger batch.
Understanding combinations is fundamental for analyzing scenarios where selection without regard to order is crucial.
