Quant
Trigonometry
Every CAT Trigonometry formula on one page — ratios, identities, height and distance, and the law of sines and cosines — each with a worked example.
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Permutations & Combinations
CAT'26 QUANT CHEATSHEET
Every permutations & combinations formula you need for CAT 2026 — on one page.
Permutations and Combinations on CAT is really a small set of ideas — ordered vs unordered, AND vs OR, together vs at-least-one — recombined into questions that look unfamiliar. Once you can spot which version of the counting problem you’re looking at, the formula is usually a direct substitution. This sheet lays out every formula you need for the CAT quant section: the fundamental counting principle, permutations and combinations and how they relate, circular arrangements, repeated-item and restricted arrangements, group division, stars and bars, derangements, and word ranking, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Permutations & Combinations: every formula you need
1Fundamental Principle of Counting
Multiply for AND, add for OR when combining independent choices.
task A (m ways) AND task B (n ways) → m×n | A OR B → m+n
Example: 3 shirts and 4 pants → outfits = 3·4 = 12.
CAT Hack: AND means multiply, OR means add — identify which one the question is really asking before counting.
2Factorial Basics
The product of all positive integers up to n.
n! = n×(n−1)×…×1 | 0! = 1
Example: 5! = 5·4·3·2·1 = 120.
3Permutations (nPr)
The number of ordered arrangements of r items chosen from n.
ⁿPr = n! / (n−r)!
Example: ⁵P3 = 5!/2! = 120/2 = 60.
4Combinations (nCr)
The number of unordered selections of r items chosen from n.
ⁿCr = n! / (r!(n−r)!)
Example: ⁵C3 = 5!/(3!2!) = 120/12 = 10.
Common Mistake: Combinations divide out the internal order; permutations keep it. Confusing the two is the single most common P&C error.
5Relationship Between nPr and nCr
A permutation is a combination followed by ordering the chosen items.
ⁿPr = ⁿCr × r!
Example: ⁵P3 = ⁵C3 × 3! = 10·6 = 60.
6Circular Permutations
Arranging objects around a circle removes one degree of freedom.
n distinct objects in a circle: (n−1)! | if clockwise=anticlockwise: (n−1)!/2
Example: 5 people at a round table → (5−1)! = 24 arrangements.
CAT Hack: Fix one object to kill the rotational symmetry, then arrange the rest in a line — that's exactly what (n−1)! does.
7Permutations with Repetition Allowed
Each position is filled independently from the full set of choices.
nr arrangements, when repetition is allowed at each of r positions
Example: 3-digit codes from digits 0–9, repetition allowed → 103 = 1000.
8Permutations of Items Not All Distinct
Divide out the internal orderings of each group of identical items.
n!/(p!q!r!…), where p, q, r… are the sizes of each identical group
Example: arrangements of BANANA (A×3, N×2) → 6!/(3!2!) = 720/12 = 60.
Common Mistake: Forgetting to divide by each repeated letter's factorial overcounts identical-looking arrangements.
9Permutations with Restrictions
Bundle items that must stay together into a single unit first.
“together”: treat the group as 1 unit, arrange, then arrange within the unit
Example: 5 people, 2 must sit together → 4! · 2! = 24·2 = 48.
10Combinations with Restrictions
“At least one” is almost always faster via the complement.
at least 1 of type X = total selections − selections with none of X
Example: choose 3 from 5 men & 4 women, at least 1 woman → C(9,3)−C(5,3) = 84−10 = 74.
CAT Favourite: The complement trick (total minus the unwanted case) turns a messy multi-case count into one subtraction.
11Dividing Items into Groups
Splitting distinct items into groups, with an extra correction if the groups are equal-sized.
distinct group sizes: n!/(p!q!r!…) | k equal-sized groups: divide further by k!
Example: 6 distinct books into 2 unlabeled groups of 3 → 6!/(3!3!2!) = 10.
CAT Insight: The extra k! only applies when the groups themselves are indistinguishable (unlabeled), not when they're assigned to different people or shelves.
12Distributing Identical Items (Stars & Bars)
Split n identical items among r distinct groups, zero allowed in a group.
number of ways = C(n+r−1, r−1)
Example: 10 identical candies among 3 children → C(12,2) = 66.
13Derangements
Permutations where no item ends up in its original position.
D(n) = n! × [1−1/1!+1/2!−1/3!+…+(−1)ⁿ/n!]
Example: D(4) = 24·(1−1+1/2−1/6+1/24) = 9.
14Rank of a Word (Dictionary Order)
Count how many alphabetically-earlier arrangements exist, then add one.
for each letter position, count earlier letters × (remaining letters)!, then sum and add 1
Example: rank of RACE among permutations of A,C,E,R → 18 words come before it → rank = 19.
CAT Hack: Work letter by letter left to right, fixing each chosen letter and counting only the smaller unused letters at each step.
15Pascal's Identity & nCr Properties
Key identities connecting combinations across different values of n and r.
ⁿCr = ⁿCn−r | ⁿCr+ⁿCr−1 = ⁿ⁺¹Cr | ΣⁿCr = 2n
Example: ⁵C2+⁵C3 = 10+10 = 20 = ⁶C₃.
CAT exam shortcuts, traps & revision
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CAT Exam Shortcuts
- ⁿPr=n!/(n−r)!; ⁿCr=n!/(r!(n−r)!); ⁿPr=ⁿCr×r!
- Circular arrangement: (n−1)!, halve it if clockwise and anticlockwise are the same
- “Must be together”: bundle as one unit, then multiply by internal arrangements
- “At least one”: use the complement (total minus none), not direct casework
- Identical items into distinct groups: stars and bars, C(n+r−1, r−1)
- Pascal's identity: ⁿCr+ⁿCr−1=ⁿ⁺¹Cr
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Most Common CAT Traps
- Using a permutation formula when order doesn't matter (or vice versa).
- Forgetting the (n−1)! adjustment for circular arrangements and using n! instead.
- Not dividing by the factorial of each repeated item's count in a multiset arrangement.
- Computing “at least one” by direct casework instead of the faster complement method.
- Dividing by an extra k! for group splits even when the resulting groups are distinguishable.
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30-Second Revision Box
- ⁿPr=n!/(n−r)!; ⁿCr=n!/(r!(n−r)!)
- Circular: (n−1)!, or (n−1)!/2 if flips are identical
- Repetition allowed: nᵉ; not-all-distinct: n!/(p!q!r!…)
- Groups: distinct sizes need no extra division; k equal-sized groups divide by k!
- Stars and bars: C(n+r−1, r−1)
- Derangement D(n)=n!Σ(−1)ᵀ/k!; Pascal's: ⁿCr+ⁿCr−1=ⁿ⁺¹Cr
This topic rewards identifying the right counting pattern over memorising every case separately — once you see whether order matters, whether items repeat, or whether a complement is faster than direct casework, the count falls out in one line. Drill this sheet until the circular-permutation and stars-and-bars formulas are reflex, then test them on full sets and track progress with the CAT score predictor. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
