Binomial TheoremeasyPYQ · JEE Main · 2 Apr 2026 · Shift 2 (Afternoon)Free

Vandermonde Identity Binomial Sum: m = 33 | JEE Main 2026

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

Question
If for 3≤r≤303\le r\le 30,
30C30−r+3 30C31−r+3 30C32−r+30C33−r=mCr,{}^{30}C_{30-r}+3\,{}^{30}C_{31-r}+3\,{}^{30}C_{32-r}+{}^{30}C_{33-r}={}^{m}C_r,
 then mm equals
A3131
B3232
C3333correct
D3434
Solution
Step 1: The coefficients 1,3,3,11,3,3,1 are the binomial coefficients 3C3,3C2,3C1,3C0{}^3C_3,{}^3C_2,{}^3C_1,{}^3C_0. Rewrite the left side as
3C3 30C30−r+3C2 30C31−r+3C1 30C32−r+3C0 30C33−r.{}^3C_3\,{}^{30}C_{30-r}+{}^3C_2\,{}^{30}C_{31-r}+{}^3C_1\,{}^{30}C_{32-r}+{}^3C_0\,{}^{30}C_{33-r}.
 Step 2: This is exactly the Vandermonde convolution ∑k3Ck 30C(33−r)−k\displaystyle\sum_{k}{}^3C_k\,{}^{30}C_{(33-r)-k}, which equals
33C33−r.{}^{33}C_{33-r}.
 Step 3: Using 33C33−r=33Cr{}^{33}C_{33-r}={}^{33}C_{r}, compare with mCr{}^{m}C_r:
33Cr=mCr ⇒ m=33.{}^{33}C_r={}^{m}C_r\ \Rightarrow\ m=33.
 Correct answer: (3)
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