Combinations Formula – nCr and Applications

Definition

A combination is a selection of items from a larger set where the order of selection does not matter. It is used to count the number of ways to choose m elements from a total of n distinct elements.

Combinations represent the number of ways to choose r objects from a set of n objects where the order of selection does not matter. This fundamental concept in combinatorics helps solve problems involving selection, probability, and counting without regard to arrangement.

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Definition of Combinations

The number of combinations of n objects taken r at a time is denoted and calculated as:

\[ C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

\[ ^nC_r = \frac{n!}{r!(n-r)!} \quad \text{(Alternative notation)} \]

\[ \text{where } 0 \leq r \leq n \text{ and } n, r \in \mathbb{N}_0 \]

\[ \text{Order does not matter: selecting {a,b,c} = selecting {c,a,b}} \]

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Basic Examples

Simple examples illustrating combination calculations:

\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!} = \frac{120}{2 \cdot 6} = 10 \]

\[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4!}{3! \cdot 1!} = \frac{24}{6 \cdot 1} = 4 \]

\[ \binom{6}{0} = \frac{6!}{0!(6-0)!} = \frac{6!}{1 \cdot 6!} = 1 \]

\[ \binom{n}{n} = \frac{n!}{n!(n-n)!} = \frac{n!}{n! \cdot 0!} = 1 \]

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Fundamental Properties

Essential properties governing combination operations:

\[ \binom{n}{r} = \binom{n}{n-r} \quad \text{(Symmetry Property)} \]

\[ \binom{n}{0} = \binom{n}{n} = 1 \quad \text{(Boundary Cases)} \]

\[ \binom{n}{1} = \binom{n}{n-1} = n \quad \text{(Single Selection)} \]

\[ \binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r} \quad \text{(Pascal's Identity)} \]

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Pascal's Triangle

The triangular arrangement showing combination values:

\[ \begin{array}{ccccccc} & & & \binom{0}{0} & & & \\ & & \binom{1}{0} & & \binom{1}{1} & & \\ & \binom{2}{0} & & \binom{2}{1} & & \binom{2}{2} & \\ \binom{3}{0} & & \binom{3}{1} & & \binom{3}{2} & & \binom{3}{3} \end{array} \]

\[ = \begin{array}{ccccccc} & & & 1 & & & \\ & & 1 & & 1 & & \\ & 1 & & 2 & & 1 & \\ 1 & & 3 & & 3 & & 1 \end{array} \]

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Important Identities

Key identities involving combinations:

\[ \sum_{r=0}^{n} \binom{n}{r} = 2^n \quad \text{(Binomial Theorem Sum)} \]

\[ \sum_{r=0}^{n} \binom{n}{r} x^r = (1+x)^n \quad \text{(Binomial Expansion)} \]

\[ \binom{n}{r} = \frac{n}{r} \binom{n-1}{r-1} \quad \text{(Recursive Formula)} \]

\[ \sum_{k=r}^{n} \binom{k}{r} = \binom{n+1}{r+1} \quad \text{(Hockey Stick Identity)} \]

🎯 What does this mean?

Combinations answer the question "In how many ways can I choose?" when order doesn't matter. Think of it like selecting a committee from a group - whether you pick Alice then Bob, or Bob then Alice, you get the same committee. Combinations count unique selections, not arrangements.

\[ n \]

Total Objects - Total number of items available for selection

\[ r \]

Objects Selected - Number of items chosen from the total

\[ \binom{n}{r} \]

Combination Symbol - "n choose r" or number of ways to select r from n

\[ C(n,r) \]

Function Notation - Alternative way to write combinations

\[ ^nC_r \]

Legacy Notation - Older mathematical notation for combinations

\[ n! \]

Factorial - Product of all positive integers from 1 to n

\[ r! \]

Selection Factorial - Accounts for arrangements within selected group

\[ (n-r)! \]

Remainder Factorial - Accounts for unselected objects

\[ 0! \]

Zero Factorial - Defined as 1 by convention

\[ 2^n \]

Total Subsets - Sum of all possible combinations from set of n elements

\[ x^r \]

Binomial Term - Variable term in binomial expansion

\[ \mathbb{N}_0 \]

Natural Numbers with Zero - Non-negative integers {0,1,2,3,...}

🎯 Essential Insight: Combinations = Permutations ÷ Arrangements! Since order doesn't matter, we divide out the r! ways to arrange the selected items. Remember: choosing ≠ arranging! 🔄

🚀 Real-World Applications

🎲 Probability & Statistics

Lottery, Card Games & Risk Analysis

Statisticians use combinations to calculate probabilities in lotteries, poker hands, survey sampling, and medical trial designs

💼 Business & Management

Team Formation & Resource Allocation

Managers use combinations to determine team compositions, project assignments, and optimal resource distribution strategies

🧬 Genetics & Biology

DNA Analysis & Breeding Programs

Biologists apply combinations to analyze genetic variations, calculate inheritance probabilities, and design breeding experiments

💻 Computer Science & Technology

Algorithm Design & Network Security

Programmers use combinations for encryption algorithms, database queries, network routing, and optimization problems

The Magic: Probability: Total outcomes → Favorable selections, Business: Available people → Optimal teams, Genetics: Gene pools → Inheritance patterns, Technology: Data elements → Efficient algorithms

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Master the "Selection vs Arrangement" Distinction!

Before diving into formulas, understand this fundamental concept:

Key Insight: Combinations are about CHOOSING, not ARRANGING. Think "committee selection" not "seating arrangement" - the team {Alice, Bob, Carol} is the same regardless of who you picked first!

💡 Why this matters:

🔋 Real-World Power:

Lottery Systems: Calculate exact odds of winning by selecting correct number combinations
Medical Trials: Determine how many ways to select patient groups for controlled studies
Investment Planning: Count portfolio combinations when diversifying across asset classes
Quality Control: Calculate sampling strategies for product testing and inspection

🧠 Mathematical Insight:

Combinations eliminate redundant counting due to order differences
Pascal's triangle reveals beautiful patterns and recursive relationships
Binomial theorem connects combinations to algebraic expansions

🚀 Practice Strategy:

1 Identify Selection vs Arrangement 🤔

Ask: "Does order matter for this problem?"
Selection (Combinations): Teams, committees, groups, sets
Arrangement (Permutations): Passwords, races, sequences, rankings

2 Use the Formula Systematically 📋

Write: C(n,r) = n! / (r!(n-r)!)
Calculate: Work step-by-step, simplify before multiplying
Check: Use symmetry property C(n,r) = C(n,n-r) to verify

3 Leverage Pascal's Triangle Patterns 🔺

Small values: Memorize first few rows for quick calculation
Pascal's Identity: C(n,r) = C(n-1,r-1) + C(n-1,r)
Recognize patterns: Each number is sum of two above it

4 Apply Key Properties 🔗

Symmetry: C(n,r) = C(n,n-r) - choosing r = not choosing (n-r)
Boundary: C(n,0) = C(n,n) = 1 - one way to choose all or none
Sum: All combinations from n objects = 2^n total subsets

When you see combinations as the mathematical way to count "pure selections" without caring about order, combinatorics becomes a powerful tool for solving real-world problems involving choice and probability!

Memory Trick: "Combinations = Committees" - CHOOSING: Select members for a group, ORDER: Doesn't matter who you pick first, FORMULA: Divide out arrangements (r!)

🔑 Key Properties of Combinations

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Symmetry Property

Choosing r objects = not choosing (n-r): C(n,r) = C(n,n-r)

Selecting is equivalent to leaving behind

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Pascal's Identity

Recursive relationship: C(n,r) = C(n-1,r-1) + C(n-1,r)

Each combination equals sum of two combinations above it

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Boundary Values

Special cases: C(n,0) = C(n,n) = 1, C(n,1) = C(n,n-1) = n

One way to choose everything/nothing, n ways to choose one

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Binomial Sum

Sum of all combinations: Σ C(n,r) = 2^n for r from 0 to n

Total number of subsets of n-element set

Universal Insight: Combinations are the mathematical foundation of choice - they quantify possibilities when order is irrelevant, making them essential for probability, statistics, and optimization! 🎯

Order Independence: {A,B,C} = {C,A,B} = {B,C,A} - all count as one combination

Symmetry Rule: C(10,3) = C(10,7) - choosing 3 = leaving 7

Pascal's Pattern: Each value equals sum of two values above in triangle

Total Count: Sum of all C(n,r) values = 2^n possible subsets

Combinations – Selecting Items Without Order