Binomial TheoremmediumPYQ · JEE Main · 2 Apr 2026 · Shift 1 (Morning)Free

Binomial Identity Set with Integer k: 4 Elements | JEE Main 2026

JEE Maths question with a full step-by-step solution.

Question

The number of elements in the set

S=\left\{(r,k):k\in\mathbb{Z}\ \text{and}\ {}^{36}C_{r+1}=\dfrac{6\left({}^{35}C_r\right)}{k^2-3}\right\}

2

4

correct

8

16

Solution

Step 1: Use

{}^{36}C_{r+1}=\dfrac{36}{r+1}\,{}^{35}C_r

. Substitute:

\frac{36}{r+1}\,{}^{35}C_r=\frac{6\,{}^{35}C_r}{k^2-3}\ \Rightarrow\ \frac{36}{r+1}=\frac{6}{k^2-3}.

Step 2: Cross-multiply:

k^2-3=\frac{6(r+1)}{36}=\frac{r+1}{6}\ \Rightarrow\ k^2=\frac{r+19}{6}.

Step 3: Constraint

0\le r\le 35

gives

\frac{0+19}{6}\le k^2\le\frac{35+19}{6}\ \Rightarrow\ 3.16\ldots\le k^2\le 9.

So the perfect-square integer values are

k^2=4

and

k^2=9

. Step 4:

k^2=4\Rightarrow r+19=24\Rightarrow r=5

, with

k=\pm2

k^2=9\Rightarrow r+19=54\Rightarrow r=35

, with

k=\pm3

. Step 5: The valid pairs are

(5,2),(5,-2),(35,3),(35,-3)

— that is

4

elements. Correct answer: (2)

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