Combinations
Number of unordered selections of r items from n.
Formula
Variables
Example
C(5,2) = 10
How it works
Combinations count how many ways you can choose a group of r items from n items when order does not matter. The factorial formula works because it starts with all possible ordered arrangements of r chosen items from n, then removes the extra orderings by dividing by r!, and also accounts for the unchosen items through the (n-r)! term. This is why C(n,r) gives a count of distinct groups rather than sequences.
When to use it
Use this when selecting a committee, picking lottery numbers, choosing toppings, or forming any group where the arrangement is irrelevant. It is the right tool whenever you only care about which items are chosen, not the order they are picked in.
Worked example, step by step
Find the number of ways to choose 3 students from 7. C(7,3) = 7! / (3!(7-3)!) C(7,3) = 7! / (3!4!) C(7,3) = (7 × 6 × 5 × 4!) / ((3 × 2 × 1) × 4!) C(7,3) = (7 × 6 × 5) / 6 C(7,3) = 35 So, there are 35 ways to choose 3 students from 7.
Common mistakes to avoid
- Using combinations when the order of selection matters
- Forgetting to subtract r from n in the denominator term (n-r)!
- Calculating n! and r! separately without simplifying, which often leads to unnecessary arithmetic errors
Did You Know?
The binomial coefficient counts subsets and appears in Pascal triangle.
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