JEE Main Maths · Binomial Theorem

Binomial Coefficients in JEE Main

Name: Binomial Coefficients: JEE Main previous year questions
Creator: Lumi

Binomial Coefficients (Maths, Binomial Theorem) came 26 times in 26 of the 42 JEE Main shifts Lumi analysed, across 18 distinct ideas. 17 of them came in 2024–26; it was last asked in 2026.

Questions: 26; in 26 shifts
Ideas asked: 18; 6 asked more than once
Numerical answer: 13; 50% of its questions
Pattern: Asked most years; last asked 2026

Questions each year

In the shifts Lumi analysed for each year.

Difficulty and format

Easy 4Medium 12Hard 10

Format

Single correct13 Q
Numerical value13 Q

Ideas inside the topic

18 ideas

Sum of all Coefficients3 Q
Maximum Binomial Coefficient3 Q
Sum of Even and Odd Coefficients2 Q
Second Derivative Method for Sums2 Q
Basic nCr Properties2 Q
Integration Method for Sums2 Q
Geometric Progression of Expansions1 Q
GP-Weighted Coefficient Sums1 Q
Hockey Stick Identity1 Q
Alternating Binomial Series1 Q
Differentiation Method for Sums1 Q
Maximum Coefficient for Odd n1 Q

Where it sits

Part of Binomial Theorem, which had 54 questions in all. Binomial Coefficients accounts for 48% of them. In Lumi's JEE taxonomy it belongs to Algebra.

Every question

All 26 Binomial Coefficients questions

Newest first, each with its options and official answer.

2 Apr 2026, Shift 1 · Q7The number of elements in the set…MediumSingle correct
2 Apr 2026, Shift 2 · Q6If for $3 \le r \le 30$ , $\binom{30}{C_{30-r}} + 3\binom{30}{C_{31-r}} + 3\binom{30}{C_{32-r}} + \binom{30}{C_{33-r}} = ^m C_r$ , then $m$ …MediumSingle correct
4 Apr 2026, Shift 1 · Q9Let the smallest value of $k\in\mathbb{N}$ , for which the coefficient of $x^3$ in…MediumSingle correct
5 Apr 2026, Shift 1 · Q23If the sum of the coefficients of $x^7$ and $x^{14}$ in the expansion of $\left(\frac{1}{x^3} - x^4\right)^n$ , $x \neq 0$ , is zero, then…MediumNumerical value
6 Apr 2026, Shift 1 · Q8If the coefficients of the middle terms in the binomial expansions of $(1+\alpha x)^{26}$ and $(1-\alpha x)^{28}$ , $\alpha\neq 0$ , are…EasySingle correct
8 Apr 2026, Shift 2 · Q8If…HardSingle correct
2 Apr 2025, Shift 2 · Q7If $\sum_{r=0}^{10}\left(\frac{10^{r+1}-1}{10^r}\right)\cdot {}^{11}C_{r+1} = \frac{\alpha^{11}-11^{11}}{10^{10}}$ , then $\alpha$ is equal…HardSingle correct
3 Apr 2025, Shift 1 · Q8If $\sum_{r=1}^{9}\left(\frac{r+3}{2^r}\right)\cdot {}^9C_r = \alpha\left(\frac{3}{2}\right)^9 - \beta$ , $\alpha, \beta \in \mathbb{N}$ ,…MediumSingle correct
3 Apr 2025, Shift 2 · Q21Let $(1 + x + x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \ldots + a_{20} x^{20}$ . If $(a_1 + a_3 + a_5 + \ldots + a_{19}) - 11a_2 = 121k$ , then…HardNumerical value
4 Apr 2025, Shift 2 · Q7If $1^2\cdot({}^{15}C_1)+2^2\cdot({}^{15}C_2)+3^2\cdot({}^{15}C_3)+\ldots+15^2\cdot({}^{15}C_{15})=2^m\cdot3^n\cdot5^k$ , where…MediumSingle correct
7 Apr 2025, Shift 2 · Q21The sum of the series…HardNumerical value
8 Apr 2024, Shift 1 · Q24Let $\alpha=\sum_{r=0}^{n}(4r^2+2r+1)\,{}^nC_r$ and $\beta=\left(\sum_{r=0}^{n}\frac{{}^nC_r}{r+1}\right)+\frac{1}{n+1}$ . If…HardNumerical value
9 Apr 2024, Shift 1 · Q4The coefficient of $x^{70}$ in $x^2(1+x)^{98}+x^3(1+x)^{97}+x^4(1+x)^{96}+\ldots+x^{54}(1+x)^{46}$ is ${}^{99}C_p-{}^{46}C_q$ . Then a…HardSingle correct
29 Jan 2024, Shift 1 · Q23If $\frac{{}^{11}C_1}{2}+\frac{{}^{11}C_2}{3}+\ldots+\frac{{}^{11}C_9}{10}=\frac{n}{m}$ with $\gcd(n,m)=1$ , then $n+m$ is equal to ______.MediumNumerical value
30 Jan 2024, Shift 2 · Q24Let $\alpha = \sum_{k=0}^{n} \left(\frac{({}^nC_k)^2}{k+1}\right)$ and…HardNumerical value
31 Jan 2024, Shift 1 · Q24In the expansion of $(1+x)(1-x^2)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5,\ x\neq0$ , the sum of the coefficients of $x^3$ …MediumNumerical value
31 Jan 2024, Shift 2 · Q23Let the coefficient of $x^r$ in the expansion of $(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots+(x+2)^{n-1}$ be $\alpha_r$ . If…HardNumerical value
6 Apr 2023, Shift 1 · Q5If ${}^{2n}C_3 : {}^{n}C_3 = 10:1$ , then the ratio $(n^2+3n):(n^2-3n+4)$ isEasySingle correct
11 Apr 2023, Shift 1 · Q21The mean of the coefficients of $x, x^2, \ldots, x^7$ in the binomial expansion of $(2+x)^9$ is ______________.MediumNumerical value
25 Jul 2022, Shift 1 · Q23If the maximum value of the term independent of $t$ in the expansion of…HardNumerical value
25 Jul 2021, Shift 1 · Q83The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in…EasyNumerical value
16 Mar 2021, Shift 2 · Q87Let n be a positive integer. Let…MediumNumerical value
18 Mar 2021, Shift 1 · Q66Let $(1+x+2x^2)^{20} = a_0 + a_1x + a_2x^2 + \ldots + a_{40}x^{40}$ . Then, $a_1 + a_3 + a_5 + \ldots + a_{37}$ is equal to :HardSingle correct
8 Jan 2020, Shift 1 · Q55If a, b and c are the greatest values of $^{19}C_p$ , $^{20}C_q$ and $^{21}C_r$ respectively, then :EasySingle correct
9 Jan 2020, Shift 2 · Q71If $C_r\equiv{}^{25}C_r$ and $C_0+5\cdot C_1+9\cdot C_2+\ldots+(101)\cdot C_{25}=2^{25}\cdot k$ , then $k$ is equal to ______.MediumNumerical value
2 Apr 2017 (offline) · Q67The value of $(^{21}C_1-{}^{10}C_1)+(^{21}C_2-{}^{10}C_2)+(^{21}C_3-{}^{10}C_3)+(^{21}C_4-{}^{10}C_4)+\ldots+(^{21}C_{10}-{}^{10}C_{10})$ …MediumSingle correct