Permutation & Combination - Revision Notes
Permutation and Combination are fundamental counting techniques in mathematics that help us determine the number of ways to arrange or select objects. These concepts are crucial for solving probability problems and are frequently tested in major Philippine entrance exams like UPCAT, CSE, and NMAT. Understanding when to use permutation (order matters) versus combination (order doesn't matter) is key to mastering this topic.
Understanding Permutation
Formulas
nPr = n!/(n-r)!
Example
How many ways can 3 students be arranged in a row from 5 students? 5P3 = 5!/(5-3)! = 5!/2! = 120/2 = 60 ways
Variables
n = total number of objects, r = number of objects to arrange, ! = factorial
Application
Finding number of ways to arrange r objects from n total objects where order matters
Exam tips
- Look for keywords: arrange, order, position, sequence, ranking
- If the problem asks 'in how many ways can objects be arranged', use permutation
- Practice identifying when order matters vs when it doesn't
Key points
- Permutation is used when the ORDER of arrangement MATTERS
- Different arrangements of the same objects are counted as different permutations
- Example: Arranging letters A, B, C gives ABC, ACB, BAC, BCA, CAB, CBA - all different
- Used in situations involving positions, rankings, or sequences
- Formula involves n! (factorial) in numerator and (n-r)! in denominator
Definitions
Factorial
Product of all positive integers less than or equal to a given number. Example: 4! = 4 × 3 × 2 × 1 = 24
Importance
Essential for calculating both permutations and combinations
Permutation
An arrangement of objects where the order of selection matters
Importance
Used when position or sequence is important in the problem
Common mistakes
- Confusing permutation with combination - remember permutation cares about ORDER
- Forgetting to subtract r from n in the denominator factorial
- Not recognizing when order matters in word problems
Understanding Combination
Formulas
nCr = n!/[r!(n-r)!]
Example
How many ways can 3 artists be chosen from 10 talented artists? 10C3 = 10!/(3!×7!) = (10×9×8)/(3×2×1) = 720/6 = 120 ways
Variables
n = total number of objects, r = number of objects to select, ! = factorial
Application
Finding number of ways to select r objects from n total objects where order doesn't matter
Exam tips
- Look for keywords: choose, select, pick, combination, group
- If BC and CB are considered the same, use combination
- Remember: nCr = nC(n-r), which can simplify calculations
Key points
- Combination is used when the ORDER of selection does NOT matter
- Different arrangements of the same selected objects count as ONE combination
- Example: Selecting 2 letters from A, B, C gives AB, AC, BC - only 3 combinations (not 6)
- Used in selection problems where position doesn't matter
- Formula has additional r! in denominator to eliminate order arrangements
Definitions
Combination
A selection of objects where the order of selection does not matter
Importance
Used when choosing or selecting items without regard to arrangement
Common mistakes
- Using combination when order actually matters
- Forgetting the additional r! in the denominator
- Misidentifying selection vs arrangement problems
Problem-Solving Strategies
Formulas
nPr = nCr × r!
Example
5P3 = 5C3 × 3! = 10 × 6 = 60, while 5C3 = 10
Variables
This shows the relationship between permutation and combination
Application
Understanding why permutation values are always larger than combination values
Exam tips
- Create a mental checklist: Does order matter? Use permutation. Order doesn't matter? Use combination.
- Practice identifying keywords that indicate permutation vs combination
- Double-check your formula choice before calculating
Key points
- Always identify whether order matters (permutation) or not (combination)
- Read word problems carefully to understand what is being asked
- Practice with real-world examples like elections, committees, passwords
- Use the relationship between permutation and combination: nPr = nCr × r!
- For large numbers, look for ways to simplify calculations
Definitions
Arrangement Problem
A problem where the order or position of objects matters, requiring permutation
Importance
Recognizing this type helps choose the correct formula
Selection Problem
A problem where only the choice of objects matters, not their order, requiring combination
Importance
Distinguishing this from arrangement problems is crucial
Common mistakes
- Not reading the problem carefully to determine if order matters
- Mixing up the formulas for permutation and combination
- Calculating factorials incorrectly for large numbers
Applications and Real-World Examples
Exam tips
- Practice with Filipino context examples: barangay elections, class officers, jeepney seating
- Break complex problems into simpler parts
- Verify your answer makes logical sense in the context
Key points
- Elections: Selecting senators (combination) vs ranking candidates (permutation)
- Committees: Choosing members is combination, assigning roles is permutation
- Passwords: Arrangements with specific positions use permutation
- Sports: Team selection is combination, batting order is permutation
- Academic: Course selection is combination, class schedule is permutation
Definitions
Consecutive Numbers
Numbers that follow each other in order without gaps, like 1,2,3,4 or 23,24,25
Importance
Understanding consecutive number properties helps in certain combination problems
Common mistakes
- Assuming all selection problems use combination - some require considering position
- Not recognizing when a problem has both selection and arrangement components
- Overlooking restrictions or conditions in the problem
Connections
- Permutation and combination connect to probability theory - used to count favorable outcomes
- Factorial calculations are essential in both permutation and combination formulas
- These concepts apply to statistics, especially in sampling and experimental design
- Real-world applications include cryptography, genetics, and computer science algorithms
- Understanding these helps in advanced topics like binomial theorem and probability distributions
Quick review questions
A director wants to choose 3 artists from 10 to participate in an art class. How many ways can this be done?
Answer
120 ways
This is a combination problem since order doesn't matter in selection. Use 10C3 = 10!/(3!×7!) = (10×9×8)/(3×2×1) = 720/6 = 120
In how many ways can 5 students be arranged in a row for a photo?
Answer
120 ways
This is a permutation problem since position in the row matters. Use 5P5 = 5! = 5×4×3×2×1 = 120
A security system uses a 4-letter password where no letter can be repeated. How many passwords are possible?
Answer
358,800
This is permutation since position matters in passwords. With 26 letters available: 26P4 = 26×25×24×23 = 358,800
From 7 candidates, how many groups of 5 can be chosen for junior auditor positions?
Answer
21 ways
This is combination since we're just selecting people, not assigning specific roles. Use 7C5 = 7!/(5!×2!) = (7×6)/(2×1) = 42/2 = 21
Diagrams
Permutation and Combination Concept Overview
This mind map shows the overall structure of permutation and combination concepts, highlighting the key differences and applications of each method.
Problem-Solving Decision Process
This flowchart guides students through the decision-making process for choosing between permutation and combination, including keyword recognition and verification steps.
Formula Application Process
This flowchart demonstrates the step-by-step calculation process for both permutation and combination formulas with examples.
Exam strategy
For Philippine entrance exams, focus on quickly identifying whether a problem requires permutation or combination. Practice the standard problem types: committee selection (combination), password creation (permutation), and seating arrangements (permutation). Master the formulas and practice mental shortcuts for common calculations. Remember that combination problems often appear in probability questions, while permutation problems frequently involve sequences or rankings. Time management is crucial - spend more time identifying the problem type than on calculations, as formula errors are more common than conceptual mistakes.