Combination
A combination is a selection of objects from a set where the order of selection does not matter. The number of ways to choose r objects from n distinct objects is denoted C(n, r), nCr, or the binomial coefficient "n choose r". Combinations are used extensively in probability, statistics, the binomial theorem, and in applications such as lottery analysis, team selection, and clinical trial design.
Key Formula
C(n, r) = n! / (r! × (n − r)!)
LaTeX: C(n, r) = \binom{n}{r} = \frac{n!}{r! \, (n - r)!}
| Symbol | Meaning | Unit |
|---|---|---|
| n | Total number of distinct objects | count |
| r | Number of objects selected | count |
| n! | Factorial of n | dimensionless |
| r! | Factorial of r (accounts for order not mattering) | dimensionless |
| C(n,r) | Number of unordered selections | count |
Worked Example
Problem
A cricket team of 11 players is to be selected from a squad of 15 players. In how many ways can the team be chosen?
Solution
Step 1: Identify n and r. — n = 15 (squad size), r = 11 (team size) — Order does not matter (no specific roles assigned). Step 2: Apply the combination formula. — C(15, 11) = 15! / (11! × (15 − 11)!) = 15! / (11! × 4!) Step 3: Simplify. — C(15, 11) = C(15, 4) [by symmetry C(n,r) = C(n, n−r)] — = (15 × 14 × 13 × 12) / (4 × 3 × 2 × 1) — = 32760 / 24 = 1365
Answer
The team can be selected in 1365 different ways.
Combination vs Permutation: Key Differences
| Feature | Combination | Permutation | Example (n=5, r=2) |
|---|---|---|---|
| Order matters? | No | Yes | C=10, P=20 |
| Formula | n! / (r!(n−r)!) | n! / (n−r)! | — |
| Relationship | C(n,r) = P(n,r) / r! | P(n,r) = r! × C(n,r) | 20/2! = 10 |
| Use case | Selecting a committee | Arranging runners in a race | — |
| Notation | nCr, C(n,r), ⁿCᵣ, (n choose r) | nPr, P(n,r) | — |
| Symmetry | C(n,r) = C(n, n−r) | P(n,r) ≠ P(n, n−r) | C(5,2)=C(5,3)=10 |
Interactive Tools
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Related Terms
Permutation
A permutation is an arrangement of all or part of a set of objects in a specific order, where the order of selection matters. The number of permutations of r objects chosen from n distinct objects is denoted P(n, r) or nPr. Permutations are central to combinatorics, probability theory, and appear in computing contexts such as sorting algorithms and cryptography.
Set Theory
Set theory is the branch of mathematical logic that studies collections of objects, called sets, and the relationships between them. It provides the foundational language for nearly all of modern mathematics, defining concepts like numbers, functions, and relations in terms of sets. Developed formally by Georg Cantor in the 1870s, it underpins areas from algebra and topology to computer science and logic.
Function (mathematics)
A function is a relation that assigns to each element of a set (called the domain) exactly one element of another set (called the codomain). Denoted f: A → B, it is a rule that maps every input to a unique output without ambiguity. Functions are among the most central concepts in mathematics and appear in calculus, linear algebra, programming, and virtually every applied science.
From Latin "combinatio" (joining together), from "combinare" (to join two by two), from "com-" (together) + "bini" (two at a time). The systematic study of combinations was advanced by Blaise Pascal in the 17th century, leading to Pascal's triangle.