Key Takeaways
- Combinations count selections where order doesn't matter
- The formula is C(n,r) = n! / [r!(n-r)!]
- Also written as "n choose r" or nCr
- Always: C(n,r) = C(n, n-r) - a useful symmetry property
- Combinations are always fewer than permutations when r > 1
What Is a Combination?
A combination is a selection of items from a larger set where the order of selection does not matter. Unlike permutations, combinations focus only on which items are chosen, not the sequence in which they're picked.
For example, if you're selecting 3 people from a group of 10 to form a committee, it doesn't matter who was chosen first, second, or third - only which 3 people were selected. This is a combination problem.
C(n,r) = n! / [r! x (n-r)!]
Example: Choosing 3 People from 10
How to Calculate Combinations Step-by-Step
Identify n and r
Determine the total number of items (n) and how many you're choosing (r). For example, picking 4 cards from a deck of 52: n=52, r=4.
Calculate the Factorials
Compute n!, r!, and (n-r)!. For C(52,4): 52! is huge, 4! = 24, and 48! is also huge - but most terms cancel out.
Apply the Formula
Divide n! by (r! x (n-r)!). For C(52,4): (52x51x50x49)/(4x3x2x1) = 6,497,400/24 = 270,725 possible 4-card hands.
Combination vs. Permutation: What's the Difference?
The key difference is whether order matters:
- Combination: Order does NOT matter. "ABC" is the same as "CBA"
- Permutation: Order DOES matter. "ABC" is different from "CBA"
For example, a 3-digit PIN (like 123) is a permutation problem because 123 and 321 are different PINs. But selecting 3 lottery numbers where order doesn't matter is a combination problem.
Formula Comparison
- Permutation: P(n,r) = n! / (n-r)!
- Combination: C(n,r) = n! / [r!(n-r)!]
Notice that combinations divide by an extra r! because we're eliminating the duplicates caused by different orderings.
Real-World Combination Examples
Lottery Odds
A lottery where you pick 6 numbers from 49 has C(49,6) = 13,983,816 possible combinations. That's why winning the lottery is so rare!
Poker Hands
A 5-card poker hand from a 52-card deck: C(52,5) = 2,598,960 possible hands.
Team Selection
Choosing a 5-person basketball team from 12 players: C(12,5) = 792 possible teams.
Pizza Toppings
If a pizza shop has 10 toppings and you can choose 3: C(10,3) = 120 different 3-topping combinations.
Frequently Asked Questions
In combinations, order doesn't matter (choosing team members). In permutations, order matters (ranking or arranging items). Combinations are used when selecting a group, while permutations are used when arranging items in a sequence.
"n choose r" (written as C(n,r) or nCr) means selecting r items from a set of n items without regard to order. It's calculated using the formula n!/[r!(n-r)!].
This symmetry exists because choosing r items to include is the same as choosing (n-r) items to exclude. For example, C(10,3) = C(10,7) = 120, because selecting 3 items is equivalent to leaving out 7 items.
No, r cannot be larger than n. You cannot choose more items than exist in the set. If r > n, the combination C(n,r) = 0 (there are zero ways to do this).
Both C(n,0) and C(n,n) equal 1. There's exactly one way to choose nothing (the empty set), and exactly one way to choose everything (the complete set).
Combinations are fundamental to probability calculations. The probability of an event is often (favorable combinations)/(total combinations). For example, the probability of a specific poker hand equals 1 divided by C(52,5) = 1/2,598,960.