JEE Main · Maths · Binomial Theorem
Binomial Coefficients: JEE Main previous year questions
67 questions in the 108 JEE Main shifts Lumi analysed, in 8 of 8 years. Pattern: Asked most years. 22 came in 2024–26. Here is how it is asked and which ideas come back.
- Questions
- 67
- in 61 of 108 shifts
- Years asked
- 8/8
- last asked 2026
- Weighted estimate
- 0.6
- model; history: 0.6 a shift on average, typically 0.5–0.7
Key findings
- 01
Asked in 8 of 8 years with analysed shifts and in 61 of 108 shifts: 67 questions, #31 of 196 topics. It came in every year with analysed shifts.
- 02
52% of its questions since 2020 tested an idea no earlier year had asked; 18 repeated an idea from the year before.
- 03
The most repeated idea is Ratio of Consecutive Coefficients, asked 9 times in 9 shifts (2020, 2021, 2022, 2023, 2024), most often as “unknown parameters from given binomial terms”.
- 04
JEE Main has asked 28 of the 62 ideas in this topic; 27 have never been asked, directly or inside another question.
- 05
Mostly asked as single correct (52%); 48% need a numerical answer, 10% are easy and 28% hard.
- 06
The skill tested most is multi-step (49%), then calculation (37%).
Questions a shift, year by year
Out of the 25 Maths questions in a shift. Years differ in how many shifts Lumi analysed, so each year is scaled to one shift.
Raw count: 2017: 1 in 1 shift · 2020: 6 in 12 shifts · 2021: 17 in 24 shifts · 2022: 14 in 24 shifts · 2023: 7 in 12 shifts · 2024: 11 in 18 shifts · 2025: 5 in 8 shifts · 2026: 6 in 9 shifts.
- Up to 2023
- 0.6
- a shift, of 25
- Since 2024
- 0.6
- 0.0
- Usual gap
- 1.3 yrs
- between years asking it
- Shifts with it
- 61/108
- 56% of analysed shifts
Recent framings
- binomial sum via integration method
- equate middle term coefficients of two expansions
- numerical: n from equal coefficient condition
Revise first
- Sum of all Coefficients · asked 2017, 2022, 2023, 2024
- Ratio of Consecutive Coefficients · asked 2020, 2021, 2022, 2023, 2024
- Integration Method for Sums · asked 2023, 2024, 2026
How it is asked
Difficulty, format and skill
Across all 67 questions on Binomial Coefficients since 2017.
Difficulty
QUESTIONS- Easy7 · 10%
- Medium41 · 61%
- Hard19 · 28%
Format
SHARE- Single correct · 35 Q52%
- Numerical value · 32 Q48%
Skill tested
SHARE- Multi-step · 33 Q49%
- Calculation · 25 Q37%
- Application · 8 Q12%
- Recall · 1 Q1%
Repeats
New idea, or one asked before?
Each column is a year. Blue questions tested an idea no earlier year had asked; navy ones went back to an idea from an earlier year. 2017 is the first year, so everything in it counts as new. Heights are question counts, and Lumi analysed 1 to 24 shifts a year, so compare the split inside a column rather than heights across years.
- Idea asked for the first time
- Idea asked in an earlier year
Coverage of the topic
JEE Main has asked 28 of the 62 ideas in this topic. 27 have never been asked yet.
- Asked directly28 · 45%
- Touched inside another question7 · 11%
- Never asked27 · 44%
Ideas come from Lumi’s JEE syllabus map. A never-asked idea is not a safe skip: across all of JEE Main, 22% of questions since 2024 tested an idea no earlier year had asked.
Drill down
Chapter, idea, question
Every idea JEE Main tested in this topic, with its eight years. Open one to read the questions.
28 ideas inside
Squares are the 8 years with analysed shifts, 2017 to 2026. A number is the questions that year when there is more than one; the shade is scaled to one shift, so a year with more analysed shifts (1 to 24 a year) does not look darker. Share is of the level above.
Ideas
Every idea inside this topic
28 different ideas were tested; 15 came more than once.
unknown parameters from given binomial terms
Numerical value · 9 shifts · 5 numericalterm independent of x in expansion
Numerical value · 6 shifts · 5 numericalsum of coefficients of binomial-type series
Numerical value · 6 shifts · 4 numericalcoefficient of term in sum of binomial expansions
Single correct · 4 shifts · 1 numericalodd-coefficient sum in trinomial expansion
Numerical value · 4 shifts · 2 numericalbinomial sum via integration method
Single correct · 3 shifts · 1 numericalequate middle term coefficients of two expansions
Single correct · 3 shifts · 1 numericalsum of binomial series with multiplier
Numerical value · 3 shifts · 2 numericalevaluate sum of r^2 times binomial coefficients
Single correct · 3 shifts · 1 numericalcoefficient in geometric series of binomial expansions
Single correct · 3 shiftsn from equal coefficient condition
Numerical value · 2 shifts · 1 numericalfind n from alternating binomial sum
Numerical value · 2 shifts · 1 numericalbinomial sum ratio from given identity
Numerical value · 2 shifts · 2 numericaldouble sum of binomial coefficients
Single correct · 2 shiftssum of squares of binomial coefficients
Single correct · 2 shiftscount elements of set defined by binomial identity
Single correct · 1 shiftidentify m from combined binomial coefficients
Single correct · 1 shiftsum of series with summation general term
Numerical value · 1 shift · 1 numericalsum of series with summation general term
Single correct · 1 shiftfind n from binomial sums with reciprocals
Numerical value · 1 shift · 1 numericalsum of products of binomial coefficients
Single correct · 1 shiftsum of series with summation general term
Numerical value · 1 shift · 1 numericalbound value of expression using binomial expansion
Single correct · 1 shiftmiddle term coefficient ratio
Numerical value · 1 shift · 1 numericalevaluate sum of products of binomial coefficients
Numerical value · 1 shift · 1 numericalsum of binomial coefficient series
Single correct · 1 shiftcoefficients in sum of two binomial expansions
Single correct · 1 shiftsum of weighted binomial coefficients series
Numerical value · 1 shift · 1 numerical
| Idea | Years asked | Times | Shifts | Format | Numerical |
|---|---|---|---|---|---|
| Ratio of Consecutive Coefficientsunknown parameters from given binomial terms | 9 | 9 | Numerical valueMulti-step | 5 | |
| Coefficient Extraction from Productsterm independent of x in expansion | 6 | 6 | Numerical valueCalculation | 5 | |
| Sum of all Coefficientssum of coefficients of binomial-type series | 6 | 6 | Numerical valueMulti-step | 4 | |
| Hockey Stick Identitycoefficient of term in sum of binomial expansions | 4 | 4 | Single correctApplication | 1 | |
| Sum of Even and Odd Coefficientsodd-coefficient sum in trinomial expansion | 4 | 4 | Numerical valueMulti-step | 2 | |
| Integration Method for Sumsbinomial sum via integration method | 3 | 3 | Single correctCalculation | 1 | |
| Maximum Binomial Coefficientequate middle term coefficients of two expansions | 3 | 3 | Single correctApplication | 1 | |
| Binomial Coefficients with AP Multipliersum of binomial series with multiplier | 3 | 3 | Numerical valueCalculation | 2 | |
| Second Derivative Method for Sumsevaluate sum of r^2 times binomial coefficients | 3 | 3 | Single correctCalculation | 1 | |
| Geometric Progression of Expansionscoefficient in geometric series of binomial expansions | 3 | 3 | Single correctMulti-step | – | |
| Basic nCr Propertiesn from equal coefficient condition | 2 | 2 | Numerical valueCalculation | 1 | |
| Alternating Binomial Seriesfind n from alternating binomial sum | 2 | 2 | Numerical valueCalculation | 1 | |
| AP-Weighted Coefficient Sumsbinomial sum ratio from given identity | 2 | 2 | Numerical valueCalculation | 2 | |
| Double Summation of Coefficientsdouble sum of binomial coefficients | 2 | 2 | Single correctMulti-step | – | |
| Sum of Squares of Binomial Coefficientssum of squares of binomial coefficients | 2 | 2 | Single correctCalculation | – | |
| Absorption Identitycount elements of set defined by binomial identity | 1 | 1 | Single correctMulti-step | – | |
| Pascal's Identityidentify m from combined binomial coefficients | 1 | 1 | Single correctApplication | – | |
| Falling Factorial Polynomial Splitsum of series with summation general term | 1 | 1 | Numerical valueMulti-step | 1 | |
| GP-Weighted Coefficient Sumssum of series with summation general term | 1 | 1 | Single correctMulti-step | – | |
| Definite Integral Method for Beta-like Sumsfind n from binomial sums with reciprocals | 1 | 1 | Numerical valueMulti-step | 1 | |
| Product Sums with Constant Index Differencesum of products of binomial coefficients | 1 | 1 | Single correctMulti-step | – | |
| r-Weighted Sum of Squaressum of series with summation general term | 1 | 1 | Numerical valueCalculation | 1 | |
| Bounding Binomial Coefficientsbound value of expression using binomial expansion | 1 | 1 | Single correctApplication | – | |
| Maximum Coefficient for Odd nmiddle term coefficient ratio | 1 | 1 | Numerical valueCalculation | 1 | |
| Vandermonde's Identityevaluate sum of products of binomial coefficients | 1 | 1 | Numerical valueMulti-step | 1 | |
| Zero Alternating Momentssum of binomial coefficient series | 1 | 1 | Single correctMulti-step | – | |
| Difference of Binomial Expansionscoefficients in sum of two binomial expansions | 1 | 1 | Single correctCalculation | – | |
| Differentiation Method for Sumssum of weighted binomial coefficients series | 1 | 1 | Numerical valueCalculation | 1 |
Times and shifts count every analysed shift. Across JEE Main, 78% of questions since 2024 went back to an idea an earlier year had asked; the rest were new. All repeated ideas
Every question
Binomial Coefficients questions, year by year
The newest 40 show their opening words as they appeared in the official paper; older ones name the idea they tested. Open one for its options and the official answer.
JEE Main 20266 questions in 6 shifts
- 2 Apr, Shift 1Q7The number of elements in the set…
Idea: Absorption Identity
New ideaMediumSingle correct - 2 Apr, Shift 2Q6If for , , then …
Idea: Pascal's Identity
New ideaMediumSingle correct - 4 Apr, Shift 1Q9Let the smallest value of , for which the coefficient of in…
Idea: Hockey Stick Identity
MediumSingle correct - 5 Apr, Shift 1Q23If the sum of the coefficients of and in the expansion of , , is zero, then…
Idea: Basic nCr Properties
MediumNumerical value - 6 Apr, Shift 1Q8If the coefficients of the middle terms in the binomial expansions of and , , are…
Idea: Maximum Binomial Coefficient
EasySingle correct - 8 Apr, Shift 2Q8If…
Idea: Integration Method for Sums
HardSingle correct
JEE Main 20255 questions in 5 shifts
- 2 Apr, Shift 2Q7If , then is equal…
Idea: GP-Weighted Coefficient Sums
New ideaHardSingle correct - 3 Apr, Shift 1Q8If , ,…
Idea: Binomial Coefficients with AP Multiplier
MediumSingle correct - 3 Apr, Shift 2Q21Let . If , then…
Idea: Sum of Even and Odd Coefficients
HardNumerical value - 4 Apr, Shift 2Q7If , where…
Idea: Second Derivative Method for Sums
Asked the year beforeMediumSingle correct - 7 Apr, Shift 2Q21The sum of the series…
Idea: Falling Factorial Polynomial Split
New ideaHardNumerical value
JEE Main 202411 questions in 11 shifts
- 4 Apr, Shift 2Q4If the coefficients of , and in the expansion of are in the arithmetic progression, then the maximum value of…
Idea: Ratio of Consecutive Coefficients
Asked the year beforeMediumSingle correct - 5 Apr, Shift 1Q24If the constant term in the expansion of is , then is equal to ________.
Idea: Coefficient Extraction from Products
MediumNumerical value - 6 Apr, Shift 1Q23If the second, third and fourth terms in the expansion of are , and , respectively, then …
Idea: Ratio of Consecutive Coefficients
Asked the year beforeMediumNumerical value - 8 Apr, Shift 1Q24Let and . If…
Idea: Second Derivative Method for Sums
HardNumerical value - 9 Apr, Shift 1Q4The coefficient of in is . Then a…
Idea: Geometric Progression of Expansions
HardSingle correct - 1 Feb, Shift 1Q23If the Coefficient of in the expansion of ; is , then…
Idea: Coefficient Extraction from Products
HardNumerical value - 27 Jan, Shift 1Q6If A denotes the sum of all the coefficients in the expansion of and B denotes the sum of all the coefficients in the…
Idea: Sum of all Coefficients
Asked the year beforeEasySingle correct - 29 Jan, Shift 1Q23If with , then is equal to ______.
Idea: Integration Method for Sums
Asked the year beforeMediumNumerical value - 30 Jan, Shift 2Q24Let and…
Idea: Definite Integral Method for Beta-like Sums
New ideaHardNumerical value - 31 Jan, Shift 1Q24In the expansion of , the sum of the coefficients of …
Idea: Coefficient Extraction from Products
MediumNumerical value - 31 Jan, Shift 2Q23Let the coefficient of in the expansion of be . If…
Idea: Sum of all Coefficients
Asked the year beforeHardNumerical value
JEE Main 20237 questions in 6 shifts
- 6 Apr, Shift 1Q5If , then the ratio is
Idea: Basic nCr Properties
New ideaEasySingle correct - 8 Apr, Shift 1Q8If the coefficients of three consecutive terms in the expansion of are in the ratio , then the coefficient of the fourth…
Idea: Ratio of Consecutive Coefficients
Asked the year beforeMediumSingle correct - 11 Apr, Shift 1Q21The mean of the coefficients of in the binomial expansion of is ______________.
Idea: Sum of all Coefficients
Asked the year beforeMediumNumerical value - 11 Apr, Shift 2Q8The sum of the coefficients of three consecutive terms in the binomial expansion of , which are in the ratio , is…
Idea: Ratio of Consecutive Coefficients
Asked the year beforeMediumSingle correct - 12 Apr, Shift 1Q5The sum, of the coefficients of the first terms in the binomial expansion of , is equal to
Idea: Alternating Binomial Series
MediumSingle correct - 12 Apr, Shift 1Q6If…
Idea: Integration Method for Sums
New ideaMediumSingle correct - 13 Apr, Shift 1Q23Let be the constant term in the binomial expansion of , . If the sum of the…
Idea: Sum of all Coefficients
Asked the year beforeHardNumerical value
JEE Main 202214 questions in 12 shifts
- 25 Jul, Shift 1Q23If the maximum value of the term independent of in the expansion of…
Idea: Maximum Binomial Coefficient
HardNumerical value - 26 Jul, Shift 1Q25If the coefficients of and in the expansion of , ,…
Idea: Coefficient Extraction from Products
New ideaMediumNumerical value - 26 Jul, Shift 2Q4is equal to
Idea: Double Summation of Coefficients
Asked the year beforeMediumSingle correct - 27 Jul, Shift 2Q23Let for the term in the binomial expansion of , in the increasing powers of , to be the greatest for…
Idea: Ratio of Consecutive Coefficients
Asked the year beforeHardNumerical value - 28 Jul, Shift 2Q84If is equal to , where m is odd, then…
Idea: Sum of all Coefficients
MediumNumerical value - 29 Jul, Shift 2Q25If , then is equal to ________.
Idea: r-Weighted Sum of Squares
New ideaMediumNumerical value - 30 Jul, Shift 2Q66The coefficient of in the expression is
Idea: Geometric Progression of Expansions
New ideaMediumSingle correct - 30 Jul, Shift 2Q82If and , respectively, are the second largest and the fourth largest prime numbers in the factorization of the coefficient of…
Idea: Coefficient Extraction from Products
New ideaHardNumerical value - 25 Jun, Shift 1Q21Let denote the binomial coefficient of in the expansion of . If for ,…
Idea: AP-Weighted Coefficient Sums
Asked the year beforeHardNumerical value - 25 Jun, Shift 2Q7The coefficient of in the expression , , is
Idea: Geometric Progression of Expansions
New ideaMediumSingle correct - 25 Jun, Shift 2Q24If the sum of the co-efficients of all the positive even powers of in the binomial expansion of …
Idea: Sum of Even and Odd Coefficients
Asked the year beforeHardNumerical value - 26 Jun, Shift 2 · Q26Hockey Stick IdentityAsked the year beforeMedium
- 28 Jun, Shift 1 · Q1Product Sums with Constant Index DifferenceNew ideaHard
- 28 Jun, Shift 2 · Q4Coefficient Extraction from ProductsNew ideaMedium
JEE Main 202117 questions in 14 shifts
- 4 Aug, Shift 1 · Q65Sum of Even and Odd CoefficientsNew ideaMedium
- 4 Aug, Shift 2 · Q64Double Summation of CoefficientsNew ideaMedium
- 26 Aug, Shift 1 · Q66Second Derivative Method for SumsNew ideaMedium
- 26 Aug, Shift 2 · Q83Vandermonde's IdentityNew ideaHard
- 27 Aug, Shift 1 · Q80Sum of Squares of Binomial CoefficientsNew ideaEasy
- 24 Feb, Shift 1 · Q64Zero Alternating MomentsNew ideaMedium
- 24 Feb, Shift 2 · Q71Hockey Stick IdentityAsked the year beforeMedium
- 26 Feb, Shift 1 · Q90AP-Weighted Coefficient SumsNew ideaMedium
- 25 Jul, Shift 1 · Q83Maximum Coefficient for Odd nNew ideaEasy
- 25 Jul, Shift 2 · Q62Bounding Binomial CoefficientsNew ideaHard
- 25 Jul, Shift 2 · Q83Binomial Coefficients with AP MultiplierNew ideaMedium
- 25 Jul, Shift 2 · Q87Ratio of Consecutive CoefficientsAsked the year beforeMedium
- 16 Mar, Shift 2 · Q87Alternating Binomial SeriesNew ideaMedium
- 17 Mar, Shift 2 · Q76Sum of Squares of Binomial CoefficientsNew ideaEasy
- 17 Mar, Shift 2 · Q86Ratio of Consecutive CoefficientsAsked the year beforeMedium
- 18 Mar, Shift 1 · Q66Sum of Even and Odd CoefficientsNew ideaHard
- 18 Mar, Shift 2 · Q89Binomial Coefficients with AP MultiplierNew ideaMedium
JEE Main 20206 questions in 6 shifts
- 8 Jan, Shift 1 · Q55Maximum Binomial CoefficientNew ideaEasy
- 8 Jan, Shift 2 · Q56Difference of Binomial ExpansionsNew ideaMedium
- 9 Jan, Shift 2 · Q71Differentiation Method for SumsNew ideaMedium
- 2 Sep, Shift 2 · Q71Ratio of Consecutive CoefficientsNew ideaMedium
- 4 Sep, Shift 1 · Q55Hockey Stick IdentityNew ideaMedium
- 4 Sep, Shift 2 · Q56Ratio of Consecutive CoefficientsNew ideaMedium
JEE Main 20171 question in 1 shift
Questions from the official JEE Main Paper 1 shifts published by NTA (CBSE for 2017). Questions NTA dropped are left out. Topic and idea tags, counts and notes are Lumi’s.
More topics in Binomial Theorem
This is the only topic tagged in this chapter.