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)!}. Combinations count lottery tickets, committee selections, and any situation where you pick a group and the order of picking does not matter.
Definition
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)!}.
💡 Intuition
How many ways to choose a group? \{A, B, C\} = \{C, B, A\}.
🎯 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.
Example
Formula
Notation
C(n, r), _nC_r, or \binom{n}{r} all denote combinations of r items from n
🌟 Why It Matters
Combinations count lottery tickets, committee selections, and any situation where you pick a group and the order of picking does not matter.
💭 Hint When Stuck
Ask: does the order of selection matter? If not, count permutations first and then divide by the number of rearrangements (r!).
Formal View
Related Concepts
See Also
🚧 Common Stuck Point
C(5, 2) = C(5, 3) = 10. Choosing 2 to include = choosing 3 to exclude.
⚠️ Common Mistakes
- Using combinations when order matters — picking 1st, 2nd, and 3rd place requires permutations
- Forgetting to divide by r! when converting from permutations to combinations
- Confusing C(n, r) with C(r, n) — the larger number must be n (the pool), not r (the selection)
Go Deeper
Frequently Asked Questions
What is Combination in Math?
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)!}.
Why is Combination important?
Combinations count lottery tickets, committee selections, and any situation where you pick a group and the order of picking does not matter.
What do students usually get wrong about Combination?
C(5, 2) = C(5, 3) = 10. Choosing 2 to include = choosing 3 to exclude.
What should I learn before Combination?
Before studying Combination, you should understand: permutation, factorial.
Prerequisites
Next Steps
Cross-Subject Connections
How Combination Connects to Other Ideas
To understand combination, you should first be comfortable with permutation and factorial. Once you have a solid grasp of combination, you can move on to binomial coefficient.
Visualization
StaticVisual representation of Combination