Combinations: Counting Selections Without Order
A combination counts the number of ways to choose r items from a set of n distinct items where order does not matter. Choosing {A, B, C} is the same as {C, A, B}. The combination formula, also called the binomial coefficient, is fundamental to probability, statistics, combinatorics, and the binomial theorem. It appears in lottery odds, card game probabilities, team selection, committee formation, and countless counting problems.
The Combination Formula
C(n,r) = n! / (r! x (n-r)!)
Properties:
C(n,0) = C(n,n) = 1
C(n,1) = C(n,n-1) = n
C(n,r) = C(n, n-r) (symmetry)
C(n,r) = C(n-1,r-1) + C(n-1,r) (Pascal's rule)
Sum of row: C(n,0)+...+C(n,n) = 2^n
Relation to Permutations:
C(n,r) = P(n,r) / r!
Properties:
C(n,0) = C(n,n) = 1
C(n,1) = C(n,n-1) = n
C(n,r) = C(n, n-r) (symmetry)
C(n,r) = C(n-1,r-1) + C(n-1,r) (Pascal's rule)
Sum of row: C(n,0)+...+C(n,n) = 2^n
Relation to Permutations:
C(n,r) = P(n,r) / r!
Real-World Applications
In a lottery (pick 6 from 49), odds are 1 in C(49,6) = 13,983,816. In poker, C(52,5) = 2,598,960 possible hands. In genetics, combinations determine allele groupings. The binomial theorem uses combinations: (a+b)n = ∑ C(n,k) an-k bk.
Combination vs. Permutation
Use combinations when order does NOT matter (committees, lottery). Use permutations when order DOES matter (rankings, passwords). C(n,r) = P(n,r)/r!. From 5 people, choosing 3 for committee = C(5,3) = 10; ranking 3 positions = P(5,3) = 60.