Permutations And Combinations (Copy)
PERMUTATIONS AND COMBINATIONS — O LEVEL & IGCSE ADDITIONAL MATHEMATICS FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Factorial | n! = n(n − 1)(n − 2)…1 | Forgetting 0! = 1 |
| Zero Factorial | 0! = 1 | Writing 0 |
| Permutation | ⁿPᵣ = n!/(n − r)! | Using combination formula |
| Combination | ⁿCᵣ = n!/[r!(n − r)!] | Forgetting r! |
| Arrangement | Order matters | Treating as combination |
| Selection | Order does not matter | Treating as permutation |
| Full Arrangement | n! | Forgetting repeated restrictions |
| Repeated Objects | n!/(p!q!…) | Forgetting repeats |
| Circular Arrangement | (n − 1)! | Using n! |
| Arrangements with Restrictions | Arrange restricted items first | Ignoring restrictions |
| Grouping Together | Treat grouped items as one object | Arranging separately first |
| Objects Not Together | Total arrangements − together arrangements | Forgetting subtraction |
| Choosing r Objects | ⁿCᵣ | Using permutation |
| Arranging r Objects | ⁿPᵣ | Using combination |
| Probability with Combinations | Favourable/Total | Wrong counting method |
| Complement Method | P(A’) = 1 − P(A) | Subtracting incorrectly |
| Independent Events | Multiply probabilities | Adding instead |
| Mutually Exclusive Events | Add probabilities | Multiplying instead |
| Binomial Coefficient | ⁿCᵣ = ⁿCₙ₋ᵣ | Treating differently |
| Pascal Triangle | Used for coefficients | Wrong row |
| Arrangements of Word | Divide by repeated letters | Ignoring repeats |
| Distinguishable Arrangements | n!/(repeated factorials) | Using simple factorial |
| Seating Problems | Consider restrictions carefully | Ignoring positions |
| Committee Selection | Use combinations | Using permutations |
| Race Positions | Use permutations | Using combinations |
| Choosing and Arranging | Combination × permutation | Doing one step only |
| Replacement Problems | Depends on condition | Ignoring replacement |
| Without Replacement | Total decreases each pick | Keeping totals fixed |
| With Replacement | Total remains same | Reducing totals wrongly |
| At Least Questions | Use complement method | Counting manually incorrectly |
| At Most Questions | Add relevant cases | Missing cases |
| Exact Number Questions | Count exact case only | Mixing ranges |
| Identical Objects | Divide by repeated factorials | Treating all different |
| Leading Digit Restrictions | First position special | Ignoring zero restriction |
| Odd/Even Number Formation | Restrict last digit first | Random arrangement |
| Digit Arrangement | No repetition unless stated | Reusing digits |
| Factorial Simplification | Cancel carefully | Expanding everything unnecessarily |
| Calculator Use | Use nCr and nPr modes | Using wrong function |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK FORMULAS TABLE
| Topic | Formula |
|---|---|
| Factorial | n! |
| Permutation | ⁿPᵣ = n!/(n − r)! |
| Combination | ⁿCᵣ = n!/[r!(n − r)!] |
| Circular Arrangement | (n − 1)! |
| Repeated Arrangements | n!/(p!q!…) |
QUICK DIFFERENCE TABLE
| Situation | Use |
|---|---|
| Order matters | Permutation |
| Order does not matter | Combination |
| Arrangement around circle | Circular permutation |
| Repeated objects | Divide by repeats |
| Selection only | Combination |
COMMON FACTORIAL VALUES
| Expression | Value |
|---|---|
| 0! | 1 |
| 1! | 1 |
| 2! | 2 |
| 3! | 6 |
| 4! | 24 |
| 5! | 120 |
| 6! | 720 |
| 7! | 5040 |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
COMMON ARRANGEMENT CASES
| Situation | Method |
|---|---|
| Letters together | Treat as one object |
| Letters apart | Total − together |
| Circular seating | (n − 1)! |
| Repeated letters | Divide by repeats |
| Number formation | Restrict first digit |
QUICK PROBABILITY CONNECTIONS
| Situation | Formula |
|---|---|
| Independent Events | P(A and B) = P(A) × P(B) |
| Mutually Exclusive | P(A or B) = P(A) + P(B) |
| Complement Rule | P(A’) = 1 − P(A) |
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| Order matters | Use permutation |
| Selection only | Use combination |
| Repeated letters | Divide by factorial repeats |
| Circular arrangement | Subtract 1 from factorial |
| Together condition | Group items |
| Not together | Use complement |
| At least | Use complement method |
| Number formation | Handle first digit first |
| Probability counting | Use combinations |
| Large factorials | Cancel before multiplying |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
MOST COMMON LOSSES OF MARKS
| Mistake | Why Marks Are Lost |
|---|---|
| Using permutation instead of combination | Entire answer incorrect |
| Forgetting repeated letters | Overcounting |
| Ignoring order | Wrong counting method |
| Using n! in circle arrangements | One extra rotation counted |
| Forgetting restrictions | Invalid arrangements |
| Missing complement method | Incomplete counting |
| Wrong factorial simplification | Arithmetic errors |
| Reusing digits incorrectly | Invalid numbers |
| Ignoring first digit restriction | Impossible numbers included |
| Calculator mode mistake | Wrong nCr/nPr values |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.