Permutation and Combination is one of the most scoring and formula-driven topics in Quantitative Aptitude. But many students find them confusing because unlike simple arithmetic, here order matters sometimes, and sometimes it does not. This creates confusion about when to arrange and when to select.
If you understand these three ideas:
✔ Factorial notation
✔ Permutation (order matters)
✔ Combination (order does NOT matter)
Then you can solve ANY question effortlessly.
This Permutation and Combination guide covers all formulas, concepts, shortcuts, examples, exam tricks, and FAQs, making it the perfect one-stop resource for competitive exams.
Quick Overview: Permutation & Combination Formulas
| Concept / Situation | Considered (What Matters?) | Used (Which Value?) | Formula (With Meaning of Symbols Inside Row) |
|---|---|---|---|
| Factorial of n | Product of descending natural numbers | n! | n! = n × (n−1) × (n−2) … × 2 × 1 (n = positive integer) |
| 0 Factorial | Special case definition | 0! | 0! = 1 (convention for simplification) |
| Permutation – order matters | Arrangement | nPr | nPr = n! / (n−r)! (n = total items, r = items arranged) |
| Permutation expanded form | Ordered selection | nPr | nPr = n (n−1) … (n−r+1) (r factors) |
| Permutation of all items | Arrangement of all objects | n! | n! (arranging all n things) |
| Permutation with repeated items | Identical objects | n! / (p₁! p₂! … pᵣ!) | n = total objects, p₁, p₂… = identical counts |
| Combination – order does NOT matter | Selection only | nCr | nCr = n! / [r!(n−r)!] |
| Combination simplified | r selections | nCr | nCr = n(n−1)(n−2)… (r factors) / r! |
| Symmetry rule | Reducing calculation | nCr = nC(n−r) | r picks ↔ leftover picks |
| Special values | Boundary values | 0, n | nC0 = 1; nCn = 1 |
Formulas for Permutation & Combination
Factorial Formulas (Foundation of P&C Problems)
Factorial is the first step in solving any permutation or combination problem. Since factorials appear in almost every formula, misunderstanding them leads to repeated mistakes. So always expand or simplify factorials correctly before applying them to a question.
1. Factorial of a Positive Integer
Factorial represents the total number of ways to arrange n different items in order.
n! = n×(n−1)×(n−2)×⋯×3×2×1
This helps compute arrangements, permutations, and combinations.
2. Zero Factorial (0!)
By definition:
0!=10! = 10!=1
This is essential for formulas where r = 0 or r = n.
Why Factorial Works in P&C?
Because factorial counts arrangements, not selections.
Whenever order is involved, factorial naturally appears.
Common Mistakes
- Forgetting that 0! = 1
- Expanding factorial incorrectly
- Canceling factorials wrongly in nCr calculations
Key Tip
When simplifying factorials:
n!/(n−r)! = n(n−1)(n−2)…(n−r+1)
Use this to avoid large numbers.
Permutation Formulas (Order Matters)
A permutation is an arrangement. When the order changes, the arrangement becomes different. Even reversing the order creates a new permutation.
1. Permutation (General Case)
Number of ways to arrange r things out of n distinct things:
nPr = n!/(n−r)!
Where:
n = total items
r = items to arrange
Example
6P2 = 6×5 = 30
7P3 = 7×6×5 = 210
Why This Formula Works
Because arranging r items requires selecting and ordering them.
Each selection has r! orders, built naturally inside nPr.
Common Errors
- Using nCr instead of nPr
- Forgetting order matters
- Using factorial unnecessarily for small values
Permutation With Repetition Formula
Sometimes items repeat (like letters in MISSISSIPPI). If you treat repeated items as distinct, you overcount arrangements.
Formula
n!/p1! p2! p3!…
Where:
- p1,p2,p3… are counts of repeated items
- Sum of all pi=np_i = npi=n
Example
In the word BALLOON:
- L repeated twice
- O repeated twice
- Total letters = 7
7!/2!×2! = 5040/4 = 1260
Why We Divide
Because repeated letters create identical arrangements.
Dividing removes those duplicates.
Common Mistakes
- Forgetting to divide by all repeated groups
- Using permutation n! directly
- Miscounting identical items
Combination Formulas (Order Does NOT Matter)
A combination is a selection. Order is irrelevant. AB and BA are the same combination.
1. Combination (General Case)
Number of ways to select r items from n items:
nCr =n!/r!(n−r)!
Example
11C4 = 11×10×9×8/4×3×2×1 = 330
16C13 = 16C3 = (16×15×14)/6 = 560
Why We Divide by r!
Because permutation counts all r! arrangements of a group.
Combination removes those repeated orders:
nCr =nPr/ r!
Common Mistakes
- Treating AB and BA as different
- Forgetting the r! denominator
- Not using the shortcut nCr = nC(n−r)
2. Symmetry Rule in Combinations
nCr = nC(n−r)
Why This Works
Choosing r items
Choosing n−r items to leave out.
Example
20C18 = 20C2
Easier to compute 20C2.
Smart Tips and Practical Tricks for Solving Permutation and Combination
Mastering Permutation and Combination becomes simple when you understand how order, selection, and arrangements work together. Most students make mistakes not because formulas are difficult, but because they apply them without understanding whether order matters or not. This section breaks down the most important concepts into clear, actionable tips so you can solve questions faster and more accurately.
1. Identify Whether Order Matters First
This is the most important rule in P&C.
Before applying any formula, decide:
- If order matters → Use Permutation (nPr)
- If order does NOT matter → Use Combination (nCr)
Example:
Arranging 3 students in a line → Order matters → Permutation
Selecting 3 students for a team → Order doesn’t matter → Combination
This single step prevents most calculation errors.
2. Understand Factorial Growth Properly
Factorial values increase very quickly.
Even small mistakes in factorial calculation lead to large errors.
Example:
5! = 120
6! = 720
7! = 5040
Always check factorial values carefully before substituting them into formulas.
This helps avoid calculation mistakes in both permutations and combinations.
3. Use the nCr = nC(n − r) Shortcut to Reduce Work
Many students waste time calculating unnecessarily large factorials.
Always apply:
nCr = nC(n−r)
Example:
16C13 = 16C3
This drastically reduces calculation time and avoids large-number factorials.
4. Check if Items Repeat Before Applying Formulas
Whenever letters or objects repeat, simple permutation formulas will NOT work.
Repeated elements require:
n!/p1!p2!p3!
Example:
MISSISSIPPI has repeating letters → cannot use ordinary n!
Always check repetition first to avoid wrong answers.
5. Translate Words into Mathematical Meaning
Most P&C questions hide meaning behind words.
Learn to decode them:
- “Arrange”, “order”, “seat”, “line up” → Permutation
- “Select”, “choose”, “pick”, “form a team” → Combination
- “Codes”, “passwords”, “license plates” → Permutation
- “Committee formation”, “group selection” → Combination
Understanding the language of questions improves accuracy instantly.
6. Break Large Questions into Smaller Steps
Many exam questions combine multiple P&C concepts.
Do not try to solve everything at once.
Break into parts:
- First selection (combination)
- Then arrangement (permutation)
Example:
Select 3 people → combination
Arrange them on stage → permutation
Splitting increases clarity and reduces mistakes.
7. Practice Typical Exam Patterns
P&C questions regularly appear in:
- SSC CGL, CHSL
- Bank (IBPS, SBI)
- CAT & MBA entrance exams
- RRB Railway exams
- Campus aptitude tests
Practicing these pattern-based questions improves speed and reduces the need to memorize formulas.
FAQs About Permutation and Combination
Q1. Why is factorial important in permutation and combination?
Factorial represents the total number of ordered arrangements. Since permutations and combinations depend on arrangement count, factorial becomes the base of all formulas.
Q2. Why do we divide by r! in combinations?
Because combinations do not consider order. Permutations count ordered arrangements, so dividing by r! removes repeated arrangements that are identical.
Q3. How do I quickly decide between permutation and combination?
Check if order matters.
If yes → permutation.
If no → combination.
This rule works for every P&C question.
Q4. Why do AB and BA count differently in permutation but same in combination?
Permutation cares about order, so AB ≠ BA.
Combination ignores order, so AB = BA.
Q5. When should I use nCr = nC(n − r)?
Use it whenever r is large.
Choosing 3 out of 20 is easier than choosing 17 out of 20.
This shortcut saves time and reduces mistakes.
Q6. Why do repeated items require division by factorials?
Repeated items create many identical arrangements. Dividing by the factorials of identical items removes duplicates and gives the correct count.
Q7. Why do students commonly make mistakes in P&C?
The main reason is confusion between arrangement and selection. Misidentifying this leads to wrong formulas and incorrect answers.
Q8. Is permutation always greater than combination?
Yes, Permutation counts all arrangements, while combination counts only unique selections, so permutations are always more.
Q9. How can P&C improve exam performance?
It trains logical thinking, counting skills, and pattern recognition. These skills help across multiple quantitative topics.
Q10. What is the simplest way to master this topic?
Understand factorial, differentiate between order/no order, and practice common exam patterns. Once patterns become clear, solving becomes automatic and effortless.