What is a Combination?
A combination counts the number of ways to select r items from n items where order does NOT matter. C(n,r) is also called "n choose r" or the binomial coefficient.
C(n, r) = n! / (r! × (n − r)!)
C(10, 3) = 10! / (3! × 7!) = 120
C(10, 3) = 10! / (3! × 7!) = 120
Combinations vs Permutations
C(n,r) = P(n,r) / r! — combinations are always fewer or equal to permutations since order is ignored.
- Choosing a 3-person team from 10: C(10,3) = 120
- Lottery: picking 6 numbers from 49: C(49,6) = 13,983,816
Combinations in Pascal's Triangle
C(n,r) values form Pascal's Triangle, where each entry is the sum of the two above it:
Row 0: C(0,0) = 1
Row 1: C(1,0)=1 C(1,1)=1
Row 2: C(2,0)=1 C(2,1)=2 C(2,2)=1
Row 3: C(3,0)=1 C(3,1)=3 C(3,2)=3 C(3,3)=1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
Row 1: C(1,0)=1 C(1,1)=1
Row 2: C(2,0)=1 C(2,1)=2 C(2,2)=1
Row 3: C(3,0)=1 C(3,1)=3 C(3,2)=3 C(3,3)=1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
Pascal's triangle row n gives the coefficients in the expansion of (a+b)ⁿ (Binomial Theorem).
Key Combination Properties
C(n,0) = C(n,n) = 1 (one way to choose none or all)
C(n,1) = n (n ways to choose 1 item)
C(n,r) = C(n, n−r) (symmetry property)
C(n,r) = P(n,r) / r! (combinations are permutations without order)
C(n,1) = n (n ways to choose 1 item)
C(n,r) = C(n, n−r) (symmetry property)
C(n,r) = P(n,r) / r! (combinations are permutations without order)
The Binomial Theorem Connection
(a+b)ⁿ = Σ C(n,k) × aⁿ⁻ᵏ × bᵏ
Example: (x+1)³ = C(3,0)x³ + C(3,1)x² + C(3,2)x + C(3,3)
= x³ + 3x² + 3x + 1
Example: (x+1)³ = C(3,0)x³ + C(3,1)x² + C(3,2)x + C(3,3)
= x³ + 3x² + 3x + 1
Real-World Applications
- Lottery: UK Lotto picks 6 from 59: C(59,6) = 45,057,474 combinations
- Teams: Selecting 11 players from 16: C(16,11) = 4,368 possible teams
- Probability: Binomial probability uses C(n,r) to count successes in n trials
- Genetics: Counting gene combination possibilities