Algebra

Combinations Formula

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 ways to choose r items from n items (order doesn't matter)

nTotal number of items

rNumber of items to choose

n!n factorial (n × (n-1) × ... × 1)

🎯 Key Concepts

What you need to know about Combinations:

🎲Order Doesn't Matter

Unlike permutations, in combinations, the order of selection has no effect. {A, B} is same as {B, A}.

🔢Choosing Subsets

Used when you need to select a group or subset from a larger set without arranging them.

❗Factorial Notation

Uses n! (n factorial), which is the product of all positive integers up to n.

Problem: How many ways to choose 2 fruits from 4 (Apples, Bananas, Cherries, Dates)?

Identify n and rn = 4, r = 2

Apply FormulaC(4, 2) = 4! / (2! × (4-2)!)

Simplify FactorialsC(4, 2) = (4 × 3 × 2 × 1) / ((2 × 1) × (2 × 1))

CalculateC(4, 2) = 24 / 4 = 6

Answer: 6 different ways

Problem: A class of 10 students needs 3 representatives. How many combinations are possible?

Identify n and rn = 10, r = 3

Apply FormulaC(10, 3) = 10! / (3! × 7!)

Expand termsC(10, 3) = (10 × 9 × 8) / (3 × 2 × 1)

CalculateC(10, 3) = 720 / 6 = 120

Answer: 120 possible committees

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.

C(n, 0) is always 1. There is only one way to choose nothing.

C(n, n) is always 1. There is only one way to choose everyone.