Introduction to Combinations
Learn how to count selections where order doesn't matter using the combinations formula.
Definition
- Permutation: Order matters (arranging)
- Combination: Order doesn't matter (selecting)
- {Alice, Bob} is the same as {Bob, Alice}
- There are only 3 combinations: {A,B}, {A,C}, {B,C}
Try it now
Worked Examples
A club has 5 members. How many ways can you choose a 2-person committee?
Identify n and r
(total members), (to choose) → ,
Apply the combination formula
→ Formula set up
Calculate factorials
→
Simplify
→ ways
Answer: There are 10 different 2-person committees possible.
Common Mistakes
Confusing combinations with permutations
Why it's wrong: Permutations count arrangements (order matters), combinations count selections (order doesn't matter). but .
Correct: Ask yourself: 'Does rearranging change the outcome?' If choosing a committee, {A,B} = {B,A}, so use combinations.
Forgetting to divide by in the formula
Why it's wrong: The division by removes the duplicate orderings of the same selection.
Correct: Always use the full formula:
Calculating factorials incorrectly
Why it's wrong: Large factorials can be overwhelming. Remember that most terms cancel out.
Correct: Simplify before multiplying:
Interactive Sandbox
Expression Calculator
Try these:
History
No calculations yet
Practice Problems
16 problemsWhich formula represents the number of combinations when choosing items from items?
Why It Matters
- Lottery: How many ways can 6 numbers be drawn from 49?
- Team Selection: How many ways can you pick 5 players from 12 for a starting lineup?
- Menu Planning: How many 3-course meals from 10 dishes?
- Science: How many ways to select samples for experiments?
Real World Applications
Sports Team Selection
Coaches use combinations to understand how many different team lineups are possible.
Example:
A basketball coach choosing 5 starters from 12 players: possible lineups.
A soccer coach has 18 players and needs to choose 11 for the starting lineup.
How many different starting lineups are possible?
Step 1: Write the mathematical expression
Use :
Card Games
Combinations calculate the number of possible hands in card games.
Example:
A 5-card poker hand from 52 cards: possible hands.
How many different 7-card hands are possible in a game?
Calculate .
Step 1: Write the mathematical expression
Set up the formula:
Menu Planning
Restaurants use combinations to calculate meal options and combo deals.
Example:
Choosing 3 appetizers from 10 options: different selections.
A catering company offers 15 different dishes. A customer wants to select 4 for their event.
How many different 4-dish selections are possible?
Step 1: Write the mathematical expression
Calculate :
Key Takeaways
- 1Combinations count selections where order doesn't matter
- 2Formula: , read as 'n choose r'
- 3Key difference: Permutations = arrangements (order matters), Combinations = selections (order doesn't)
- 4: Choosing what to include equals choosing what to exclude
Frequently Asked Questions
Glossary
- Combination
- A selection of items where order does not matter
- Factorial
- The product of all positive integers up to n, written as
- Binomial coefficient
- Another name for , written as
- n choose r
- Common way to read , meaning 'the number of ways to choose r from n'
Formula Card
Combinations Formula
n = total items, r = items chosen. Symmetry: $C(n,r) = C(n, n-r)$. Special cases: $C(n,0) = C(n,n) = 1$