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Combinations Formula (nCr)
Algebra • Grades 9, 10, 11, 12
Algebra
Combinations Formula (nCr)
Master the art of selection. Learn how to calculate combinations (nCr) when the order of items doesn't matter.
The Formula
C(n,r) = n! / (r!(n-r)!)
Used for: Counting the number of unordered selections of r items chosen from a set of n items
nTotal number of items in the set (n ≥ r)
rNumber of items chosen without order
n!Factorial: n × (n-1) × ... × 1
⚡Live Interactive Quick Solver
Instant CalculationCombinations C(7, 3):35 ways
Formula: 7! / (3! × 4!) = 35
C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}
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🎯 Key Concepts
What you need to know about Combinations:
🎲
Order Doesn't MatterUnlike permutations, in combinations, the order of selection has no effect. {A, B} is same as {B, A}.
🔢
Choosing SubsetsUsed when you need to select a group or subset from a larger set without arranging them.
❗
Factorial NotationUses n! (n factorial), which is the product of all positive integers up to n.
📝 Worked Examples
Example 1: Choosing a Fruit Bowl
Problem:
How many ways to choose 2 fruits from 4 (Apples, Bananas, Cherries, Dates)?1
Identify n and r
n = 4, r = 22
Apply Formula
C(4, 2) = 4! / (2! × (4-2)!)3
Simplify Factorials
C(4, 2) = (4 × 3 × 2 × 1) / ((2 × 1) × (2 × 1))4
Calculate
C(4, 2) = 24 / 4 = 6Answer: 6 different ways
Example 2: Selecting a Committee
Problem:
A class of 10 students needs 3 representatives. How many combinations are possible?1
Identify n and r
n = 10, r = 32
Apply Formula
C(10, 3) = 10! / (3! × 7!)3
Expand terms
C(10, 3) = (10 × 9 × 8) / (3 × 2 × 1)4
Calculate
C(10, 3) = 720 / 6 = 120Answer: 120 possible committees
💡 Pro Tips
Symmetry Rule
C(n, r) is always equal to C(n, n-r). Choosing 2 people to leave is the same as choosing 8 people to stay.
Zero Case
C(n, 0) is always 1. There is only one way to choose nothing.
Choosing All
C(n, n) is always 1. There is only one way to choose everyone.
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