What is the formula for Combinations Formula (nCr)?

The mathematical formula for Combinations Formula (nCr) is C(n,r) = n! / (r!(n-r)!). It is used for counting the number of unordered selections of r items chosen from a set of n items.

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Combinations Formula (nCr)

Algebra • Grades 9, 10, 11, 12

Algebra

Combinations Formula (nCr)

Q: What do the variables in the Combinations Formula (nCr) formula represent?

In the formula C(n,r) = n! / (r!(n-r)!): n = Total number of items in the set (n ≥ r), r = Number of items chosen without order, n! = Factorial: n × (n-1) × ... × 1.

Q: How do you calculate Combinations Formula (nCr) step-by-step?

To calculate combinations formula (ncr): 1. Identify the given values (n = Total number of items in the set (n ≥ r), r = Number of items chosen without order, n! = Factorial: n × (n-1) × ... × 1). 2. Substitute the values into the formula C(n,r) = n! / (r!(n-r)!). 3. Simplify and evaluate step-by-step.

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 Calculation

Total Items (n):

Choose (r):

Combinations 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 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.

📝 Worked Examples

Example 1: Choosing a Fruit Bowl

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

Example 2: Selecting a Committee

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

💡 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.

Name: _______________________________Date: _______________Class/Period: _________

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