Combinations

Introduction to Counting Principles  

Fundamental Principle of Counting (FPC)  

If one event can happen in m ways and another can happen in n ways, then both events together can happen in m × n ways.

Example:  

Choosing a shirt (3 options) and a pair of trousers (2 options):  

Total combinations = 3 × 2 = 6 outfits.

 

Table of Content

• Permutations vs Combinations - The Core Difference
• What is a Combination?
• Factorial Notation and Rules
• Formula and Properties of Combinations
• Word Problems Involving Combinations
• Combinations in Geometry and Sets
• Combinations in Probability
• Combinations vs Permutations - Final Contrast
• Conclusion
• FAQs on Combinations

 

Permutations vs Combinations - The Core Difference

Permutation: Order matters.

Example: Choosing 2 students from A, B, C

Permutations: AB, BA, AC, CA, BC, CB → Total = 6

 

Combination: Order does not matter.

Example: Same scenario

Combinations: AB, AC, BC → Total = 3

 

What is a Combination?

Definition  

A combination is a selection of r items from a group of n items, where order does not matter.

 

Formula:  

nCr = n! / [r!(n − r)!]  

Where:  

n = total number of items  

r = number of items selected  

! = factorial notation (e.g., 4! = 4 × 3 × 2 × 1)  

 

Visual Example  

Choosing 2 students from 4: A, B, C, D

Possible combinations: AB, AC, AD, BC, BD, CD

Total combinations = ⁴C₂ = 6

 

Factorial Notation and Rules

What is Factorial (n!)?  

n! (read as "n factorial") is the product of all whole numbers up to n.

Examples:  

1. 5! = 5 × 4 × 3 × 2 × 1 = 120  

2. 3! = 3 × 2 × 1 = 6  

Special Case:  

0! = 1 (Defined by convention)

Useful Tips:  

Factorial values increase quickly. Use calculators or tables for large numbers.

When simplifying combinations:

Cancel common terms in the numerator and denominator to avoid full expansion.

 

Formula and Properties of Combinations

Standard Combination Formula:  

nCr = n! / [r!(n − r)!]

 

Symmetry Property:  

nCr = nC(n−r)

Example: 8C2 = 8C6

 

Boundary Properties:  

nC0 = 1 → Selecting nothing

nCn = 1 → Selecting all items

 

Sum Identity:  

nC0 + nC1 + nC2 + ... + nCn = 2^n

This represents the total number of subsets of a set with n elements.

 

Word Problems Involving Combinations

Simple Selection  

Choosing 3 committee members from 10 students:  

¹⁰C₃ = 120 ways  

Selecting 2 shirts from 5:  

⁵C₂ = 10 ways  

 

Conditional Selection  

Selecting 3 team members but excluding student A  

Including at least 1 female in a group of 5: Break it into cases like 1 female, 2 females, etc., and use nCr for each.

 

Common Applications:  

Cards: Selecting 4 cards from a deck: ⁵²C₄  

Teams: Forming a cricket team of 11 from 15 players: ¹⁵C₁₁  

 

Combinations in Geometry and Sets

Triangles from Polygon Vertices  

To form a triangle, choose 3 vertices from n:  

Number of triangles = nC3  

Example: In a hexagon (6 sides):  

⁶C₃ = 20 triangles  

 

Subsets  

The number of subsets of size r from a set of size n:  

nCr  

Total number of subsets (of any size):  

2^n  

 

Combinations in Probability

For Large Sample Spaces  

Example: Probability of selecting 2 heads from 4 coins (assuming heads/tails are equally likely)

Total outcomes = ⁴C₂ = 6  

Desired outcome = favorable combinations / total combinations  

 

Card Probability  

Example: Probability of drawing 2 kings from a deck of 52 cards:  

Favorable = ⁴C₂  

Total = ⁵²C₂  

Probability = ⁴C₂ / ⁵²C₂ = 6 / 1326 ≈ 0.0045  

 

Other Applications:  

• Lottery draws  

• Drawing balls from urns  

• Rolling dice with conditions  

 

Combinations vs Permutations - Final Contrast

When to Use Which:  

If order or arrangement matters → use Permutation  

If it's just selection or grouping → use Combination  

 

Examples:  

• Combination: Forming a team of 3 players from 10 → ¹⁰C₃  

• Permutation: Assigning Captain, Vice-Captain, and Bowler → ¹⁰P₃  

 

Formula Comparison:  

• Permutation:  

nPr = n! / (n − r)!

 

• Combination:  

nCr = n! / [r!(n − r)!]

 

Conclusion

combinations are important in mathematics because they help us figure out how many ways we can choose items when the order doesn't matter. From basic selections and forming committees to tackling complex probability and geometry problems, combinations offer a clear method for counting outcomes. By learning factorial notation, combination formulas, and their properties, students can tackle real-world problems and exam questions with confidence and accuracy. This makes combinations a key tool in the larger area of combinatorics and applied mathematics.

 

Related Links:

Permutation and Combination - Confused between permutation and combination? Get clarity with our interactive explanations, comparison tables, and solved problems.

Permutations and Combinations Questions - Challenge yourself with our handpicked permutations and combinations question. 

Frequently Asked Questions on Combinations

1. What is called combination?

Ans: A combination is a way of selecting items from a group, where the order does not matter. It is used when you're concerned only with which items are chosen, not how they're arranged.
Example: Choosing 2 students out of 5 for a team.

 

2. What are examples of combinations?

Ans: Examples of combinations include:

• Selecting 3 fruits from a basket of apples, bananas, and oranges → Possible combinations: AB, AC, BC

• Forming a committee of 4 people from a group of 10

• Choosing 5 lottery numbers out of 50

• Picking 2 cards from a deck without caring about the order
  
  

 

3. What is 7 combination 2?

Ans: This means selecting 2 items out of 7, denoted as ⁷C₂.
Use the formula:

nCr = n! / [r!(n − r)!]

So,

⁷C₂ = 7! / (2! × 5!) = (7 × 6) / (2 × 1) = 21

Answer: 21 combinations

 

4. How many 4-number combinations are possible?

Ans: If you're choosing 4 numbers from a set of n numbers, the number of combinations is:

nC4 = n! / [4!(n − 4)!]

For example:

• Choosing 4 numbers from 10:
  ¹⁰C₄ = 210

• Choosing 4 numbers from 49 (like in lottery):
  ⁴⁹C₄ = 211,876

 

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