Combinations
Number of ways to choose r items from n (order irrelevant).
Formula
Variables
Example
C(5,2) = 10
How it works
The combination formula counts how many different groups of size r can be formed from n items when the order inside the group does not matter. It comes from starting with all possible ordered selections, then removing overcounting: each group of r items can be arranged in r! ways, so dividing by r! fixes the repeated counting, and the remaining (n-r)! term accounts for the unused items in the factorial expression. This is why combinations are smaller than permutations whenever order is not important.
When to use it
Use this formula whenever you are selecting a subset of items and the arrangement does not change the result, such as choosing committee members, lottery numbers, cards from a deck, or toppings from a menu. It is especially useful in probability when finding the number of equally likely ways to choose items.
Worked example, step by step
Find the number of ways to choose 3 students from 8. Use the combination formula: \binom{n}{r} = \frac{n!}{r!(n-r)!} Substitute n = 8 and r = 3: \binom{8}{3} = \frac{8!}{3!(8-3)!} Simplify the factorials: \binom{8}{3} = \frac{8!}{3!5!} Cancel 5! from top and bottom: \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} Compute the value: \binom{8}{3} = \frac{336}{6} = 56 Final answer: 56 ways
Common mistakes to avoid
- Using combinations when the order matters, such as ranking or arranging people
- Forgetting to include the factorials in the denominator, which leads to overcounting
- Plugging in the wrong values for n and r, especially mixing up total items with chosen items
Did You Know?
Combinations count selections where order doesn't matter — the math behind lottery odds.
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