Permutation & Combination - Memory Anchors
Memory techniques dramatically improve recall by creating multiple pathways to the same information in your brain. When you use vivid imagery, stories, and patterns, you activate both your logical and creative thinking centers. These memory anchors will help you remember formulas, distinguish between permutations and combinations, and quickly solve problems on the CSE, UPCAT, and other major exams.
Memory anchors
Permutation vs Combination - Order Matters
Memory aid
PROM - Permutation is about Position, Ranking, Order Matters. CHOOSE - Combination is about Choosing, no Order, Only Selection Matters
Why it works
The acronyms PROM and CHOOSE help distinguish the key difference - permutations care about order (like a prom lineup), combinations just care about selection (like choosing friends)
Example usage
If the problem asks 'In how many ways can 5 students be arranged in a line?' - think PROM (order matters), use permutation
Recall trigger
When you see arrangement/ranking words, think PROM. When you see selection/choosing words, think CHOOSE
Permutation Formula nPr = n!/(n-r)!
Memory aid
Picture a PARKING lot with n spaces. You want to park r cars in specific spots. First car has n choices, second has (n-1), third has (n-2), etc. Stop after r cars, so divide by what's left (n-r)!
Why it works
Visual imagery of parking cars in specific spots reinforces that order matters and helps remember why we divide by (n-r)!
Example usage
For '5 students in 3 chairs': Picture parking 3 cars in a 5-space lot - first car has 5 spots, second has 4, third has 3. Answer: 5×4×3 = 60
Recall trigger
Think of a parking lot when you see permutation problems
Combination Formula nCr = n!/(r!(n-r)!)
Memory aid
Coach Ramirez needs to CHOOSE r players from n team members for a group photo. He has n! ways to arrange everyone, but divides by r! (arrangements within chosen group don't matter) and (n-r)! (arrangements of those not chosen don't matter)
Why it works
The story explains why we divide by both r! and (n-r)! - because internal arrangements don't matter in combinations
Example usage
For 'choose 3 senators from 15 candidates': Coach has 15! ways to arrange all, divide by 3! (chosen group arrangements) and 12! (non-chosen arrangements)
Recall trigger
Think of Coach Ramirez choosing players when you need the combination formula
Factorial n! = n × (n-1) × (n-2) × ... × 1
Memory aid
Factorial's fun, it's easy to see, multiply down from n to 1, 2, 3! Start with n, go down one by one, stop at 1 and you are done!
Why it works
Rhymes create rhythm and melody that make information stick in memory longer
Example usage
For 5!: 'Start with 5, go down one by one' = 5×4×3×2×1 = 120
Recall trigger
Sing the factorial song when computing n!
Keywords for Permutation Problems
Memory aid
ARAB - Arrange, Rank, Align, By position. These words signal permutation because they involve ORDER
Why it works
Grouping similar keywords helps quick identification of problem type during time-pressured exams
Example usage
Problem says 'arrange 7 books on a shelf' - ARAB word 'arrange' signals permutation
Recall trigger
When you see ARAB words, use permutation formulas
Keywords for Combination Problems
Memory aid
SPICE - Select, Pick, Involve, Choose, Elect. These words signal combination because order doesn't matter
Why it works
SPICE is memorable and groups the most common combination keywords together
Example usage
Problem says 'select 5 committee members' - SPICE word 'select' signals combination
Recall trigger
When you see SPICE words, use combination formulas
When n = r in Permutations
Memory aid
Like arranging all your family members for a photo - if you have 5 family members and 5 positions, it's just 5! ways to arrange everyone
Why it works
Family photo is relatable and shows why nPn = n! (everyone gets arranged)
Example usage
Arrange 6 people in 6 chairs: 6P6 = 6! = 720
Recall trigger
Think family photo when n = r
Consecutive Numbers Properties
Memory aid
Picture a ladder - each rung is exactly 1 step apart. Consecutive even numbers are like a ladder with rungs 2 steps apart (2,4,6,8). Consecutive odd numbers are also 2 steps apart (1,3,5,7)
Why it works
Visual ladder reinforces the consistent spacing between consecutive numbers
Example usage
Three consecutive numbers sum to 18: Let middle rung be n, then (n-1) + n + (n+1) = 18
Recall trigger
Think of climbing a ladder when dealing with consecutive numbers
Probability Formula P(E) = Favorable/Total
Memory aid
Probability is like your chances of getting your favorite ulam in the cafeteria - it's the number of dishes you like divided by all available dishes
Why it works
Uses familiar Filipino school experience to make abstract probability concrete
Example usage
Probability of rolling even number on dice: 3 favorable (2,4,6) ÷ 6 total = 1/2
Recall trigger
Think cafeteria choices when calculating probability
Combination Problems with Large Numbers
Memory aid
For nCr calculations, CANCEL before calculating: nCr = nC(n-r). Choose the smaller r. Example: 10C7 = 10C3 (much easier to calculate)
Why it works
Chunking breaks complex calculations into smaller, manageable pieces
Example usage
15C12 = 15C3 = (15×14×13)/(3×2×1) = 455 (much easier than calculating 15C12 directly)
Recall trigger
Always choose the smaller number when calculating nCr
Zero Factorial 0! = 1
Memory aid
Maria organized a party but 0 friends showed up. How many ways can she arrange 0 people? There's exactly 1 way - the empty arrangement! That's why 0! = 1
Why it works
Story makes the counterintuitive fact that 0! = 1 memorable and logical
Example usage
When calculating 5C5 = 5!/(5!×0!) = 120/(120×1) = 1
Recall trigger
Think of Maria's empty party when you see 0!
Multiple Events Probability P(A and B) = P(A) × P(B)
Memory aid
Picture two bridges you must cross to reach your destination. Your chance of crossing both is the probability of crossing the first bridge TIMES the probability of crossing the second bridge
Why it works
Visual metaphor of crossing bridges reinforces multiplication rule for independent events
Example usage
Probability of rolling two 6's: (1/6) × (1/6) = 1/36
Recall trigger
Think crossing two bridges for multiple events
Distinguishing Arrangement Problems
Memory aid
Walk through your classroom: (1) At the door - are we ENTERING people in specific POSITIONS? (Permutation). (2) At your desk - are we just SELECTING a GROUP without caring about positions? (Combination). (3) At the blackboard - does the problem mention ARRANGEMENTS or RANKINGS? (Permutation)
Why it works
Method of loci uses familiar spatial memory to organize decision-making process
Example usage
Students for debate team positions (door = positions matter = Permutation) vs students for science club (desk = group selection = Combination)
Recall trigger
Take a mental walk through your classroom to categorize problems
Sum of Consecutive Numbers Formula
Memory aid
Consecutive numbers are like rice grains in a line. If you have n grains, the middle grain represents the average. The sum is n × (average) = n × (first + last)/2
Why it works
Rice grains are familiar to Filipino students and visualizing the middle grain helps remember the averaging concept
Example usage
Sum of consecutive numbers 5, 6, 7: 3 × (5+7)/2 = 3 × 6 = 18
Recall trigger
Think rice grains in a line for consecutive number sums
Work Problems Formula 1/t₁ + 1/t₂ = 1/t₃
Memory aid
Juan can paint a fence alone in 6 hours (works at rate 1/6 per hour). Maria can do it in 4 hours (rate 1/4 per hour). Together, their combined rate is 1/6 + 1/4 = 1/t, where t is time working together
Why it works
Concrete story with Filipino names makes abstract work rate concept tangible
Example usage
If A finishes job in 3 hours and B in 6 hours: 1/3 + 1/6 = 1/t → t = 2 hours together
Recall trigger
Think Juan and Maria painting fence together
Circular Arrangements (n-1)!
Memory aid
Picture friends sitting around a circular table for kamayan dinner. One person's position is fixed (like 12 o'clock), so only (n-1) people need arranging. It's like a regular line with one position locked
Why it works
Familiar Filipino dining tradition makes circular permutation memorable, and fixing one position is intuitive
Example usage
Arrange 5 people around circular table: Fix one person, arrange remaining 4: (5-1)! = 4! = 24 ways
Recall trigger
Think kamayan dinner table when arranging in circles
Age Problems Setup
Memory aid
AGE problems follow pattern: Now-Past-Future table. Make columns for each person and rows for time periods. Let x = current age of one person, express others in terms of x
Why it works
Systematic table organization breaks complex age relationships into manageable pieces
Example usage
Ana is 3 times older than Ben. In 5 years, Ana will be twice as old. Table helps set up: Ana now = 3x, Ben now = x
Recall trigger
Draw a Now-Past-Future table for age problems
Card Problems - Standard Deck Facts
Memory aid
SHED the cards: 52 total, Hearts and Diamonds are red (26), Spades and Clubs are black (26), Each suit has 13 cards, 4 suits total, Deck has 4 of each rank
Why it works
SHED organizes all key facts about card decks in one memorable acronym
Example usage
Probability of drawing a Queen: 4 Queens in 52 cards = 4/52 = 1/13
Recall trigger
SHED cards to remember deck composition
Dice Problems - All Outcomes
Memory aid
Picture a dice as a small bahay kubo with 6 faces. Rolling two dice is like having two bahay kubos, giving you 6 × 6 = 36 possible combinations. Each outcome is equally likely
Why it works
Cultural reference to bahay kubo makes dice memorable, and visualization of two houses helps remember why outcomes multiply
Example usage
Probability of rolling sum of 7 with two dice: 6 ways to get 7 out of 36 total = 6/36 = 1/6
Recall trigger
Think two bahay kubos for two-dice problems
Repetition in Arrangements
Memory aid
Arranging letters with repetition is like organizing identical twins in a photo - you can't tell the twins apart, so divide by 2! for each set of twins. For n total letters with groups of identical letters, divide n! by the factorial of each group size
Why it works
Twin analogy makes clear why we divide by factorials of repeated elements
Example usage
Arrange letters in BANANA: 6 letters total, but 3 A's and 2 N's repeat. Answer: 6!/(3!×2!) = 720/12 = 60
Recall trigger
Think organizing identical twins when letters repeat
Revision game
I'm the formula you use when parking cars matters, what am I?
Answer
Permutation formula nPr
Memory link
Parking lot visual association (A2)
Coach Ramirez is choosing players and doesn't care about their positions in the photo. What formula does he use?
Answer
Combination formula nCr
Memory link
Coach Ramirez micro-story (A3)
I equal 1 even though I seem like nothing. Maria's empty party explains why. What am I?
Answer
0! = 1
Memory link
Maria's empty party story (A11)
ARAB and SPICE help you choose between two types of problems. What are they?
Answer
Permutation and Combination
Memory link
ARAB mnemonic (A5) and SPICE mnemonic (A6)
Like rice grains in a line, I help you find sums of numbers that follow each other. What am I?
Answer
Consecutive numbers formula
Memory link
Rice grains analogy (A14)
Two bahay kubos together give you 36 possibilities. What am I describing?
Answer
Rolling two dice
Memory link
Bahay kubo dice visualization (A19)
When identical twins appear in arrangements, you divide by their factorial. What concept is this?
Answer
Repetition in arrangements
Memory link
Identical twins analogy (A20)
Like crossing two bridges, multiply me to find combined probability. What rule am I?
Answer
Multiple events probability rule
Memory link
Two bridges visualization (A12)
Formula mnemonics
nPr = n!/(n-r)!
Mnemonic
Permutation: Park n cars, Remove (n-r) unused spaces
When to use
When order matters in selection (arrangements, rankings, positions)
What each part means
n = total items, r = items selected, n! = all arrangements, (n-r)! = what we don't use
nCr = n!/(r!(n-r)!)
Mnemonic
Combination: Choose from n items, Cancel arrangements in r selected and (n-r) not selected
When to use
When order doesn't matter in selection (choosing groups, committees, teams)
What each part means
n! = all arrangements, r! = arrangements within selection, (n-r)! = arrangements of non-selected
P(E) = Favorable/Total
Mnemonic
Probability: Favorable outcomes over Total outcomes
When to use
For any probability calculation with equally likely outcomes
What each part means
Favorable = outcomes we want, Total = all possible outcomes
1/t₁ + 1/t₂ = 1/t₃
Mnemonic
Work rates add up: individual Work rates combine to Total work rate
When to use
When two or more people/machines work together on the same job
What each part means
t₁, t₂ = individual completion times, t₃ = combined completion time
Quick recall chains
Steps to Solve Permutation Problems
Recall test
What are the 5 steps when you see words like 'arrange' or 'rank'?
Memory chain
A Rabbit Always Brings Carrots - Arabs Respect Amazing Brain Calculations
Items to remember
- Identify keywords (ARAB)
- Check if order matters
- Use nPr formula
- Calculate n!/(n-r)!
- Simplify the answer
Steps to Solve Combination Problems
Recall test
What are the 5 steps when you see words like 'select' or 'choose'?
Memory chain
Smart People In Class Excel - Spice Produces Incredible Culinary Excellence
Items to remember
- Identify keywords (SPICE)
- Confirm order doesn't matter
- Use nCr formula
- Calculate n!/(r!(n-r)!)
- Use shortcuts when possible
Consecutive Number Problem Steps
Recall test
What are the 5 steps to solve consecutive number problems?
Memory chain
Let X Express Equations Systematically Finding - Little Xavier Eagerly Solves Fun problems
Items to remember
- Let x = first number
- Express others as x+1, x+2, etc
- Set up equation with given condition
- Solve for x
- Find all requested numbers
Work Problem Solution Steps
Recall test
What are the 5 steps to solve work rate problems?
Memory chain
Industrious Workers Add Rates Successfully Together - I Will Add Rates Swiftly Together
Items to remember
- Identify individual work rates
- Add rates for combined work
- Set up equation 1/t₁ + 1/t₂ = 1/t₃
- Solve for unknown time
- Check answer makes sense
Age Problem Solution Method
Recall test
What are the 5 systematic steps for age word problems?
Memory chain
Never Push Forward Looking Eagerly Ahead - Nice People Find Life Experiences Amazing
Items to remember
- Create Now-Past-Future table
- Let x = one person's current age
- Express all ages in terms of x
- Set up equation from given relationship
- Solve and verify
Diagrams
Permutation and Combination Concept Map
Overview mind map showing the relationship between permutations, combinations, their formulas, keywords, and applications
Problem Identification Decision Tree
Decision flowchart to help students identify whether to use permutation or combination formulas
Problem Solving Sequence
Timeline showing the systematic approach to solving permutation and combination problems
Memory Anchor Types Distribution
Distribution of different memory anchor types used in this chapter to ensure variety and effectiveness