Combination Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Combination.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
A combination is an unordered selection of objects — the number of ways to choose r items from n distinct items is C(n,r) = \frac{n!}{r!(n-r)!}.
How many ways to choose a group? \{A, B, C\} = \{C, B, A\}.
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: Combinations count unordered groups: C(n,r) = P(n,r)/r! because the r! orderings of the same group all count as one combination.
Common stuck point: C(5, 2) = C(5, 3) = 10. Choosing 2 to include = choosing 3 to exclude.
Sense of Study hint: Ask: does the order of selection matter? If not, count permutations first and then divide by the number of rearrangements (r!).
Worked Examples
Example 1
easySolution
- 1 Recall the combination formula for unordered selections: \binom{n}{r} = \frac{n!}{r!(n-r)!}, with n = 8, r = 3.
- 2 Cancel common factorial terms: \binom{8}{3} = \frac{8!}{3! \cdot 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1}
- 3 Calculate: \frac{336}{6} = 56
Answer
Example 2
mediumPractice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
mediumExample 2
mediumRelated Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.