The number of possible combinations can be found using a specific formula that accounts for the selection of items from a larger set without regard to the order. This is different from permutations, where order matters.
Understanding Combinations
A combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. In simpler terms, if you choose items A, B, and C, it's considered the same combination as choosing C, B, and A.
The Combination Formula
The formula to calculate the number of combinations is:
*C(n, k) = n! / (k! (n - k)!)**
Where:
- n is the total number of items in the set.
- k is the number of items you are choosing from the set.
- ! represents the factorial, which is the product of all positive integers up to that number (e.g., 5! = 5 4 3 2 1).
- C(n, k) represents the number of combinations of n items taken k at a time.
Breaking Down the Formula
The formula is derived from the permutation formula. The permutation formula calculates the number of ways to arrange k items from a set of n items, where order matters. The formula is:
P(n, k) = n! / (n - k)!
To get the combination formula, we divide the number of permutations by k! because each set of k items can be arranged in k! different ways. Since we only care about the set of items and not the order we chose them, we divide by k! to avoid counting the same set multiple times.
Example
Let's say you have a set of 5 fruits (n = 5): Apple, Banana, Cherry, Date, and Elderberry. You want to choose 3 fruits (k = 3) to make a fruit salad. How many different combinations of fruits can you choose?
Using the formula:
C(5, 3) = 5! / (3! * (5 - 3)!)
C(5, 3) = 5! / (3! * 2!)
C(5, 3) = (5 4 3 2 1) / ((3 2 1) (2 1))
C(5, 3) = 120 / (6 * 2)
C(5, 3) = 120 / 12
C(5, 3) = 10
Therefore, there are 10 possible combinations of 3 fruits you can choose from a set of 5.
Summary
To find the number of possible combinations, use the formula C(n, k) = n! / (k! * (n - k)!), where n is the total number of items, and k is the number of items you are choosing. This formula ensures that you are only counting each unique group of items once, regardless of the order in which they are selected.
