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JEE Main 20262 AprilEvening Shift

Exam: JEE Main
Year: 2026
Date: 2 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2026 · 2 April · Evening Shift · Mathematics · Binomial Theorem

If for 3 ≤ r ≤ 30 , 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = m r...

If for

3 \leq r \leq 30

,

\binom{30}{30-r} + 3\binom{30}{31-r} + 3\binom{30}{32-r} + \binom{30}{33-r} = \binom{m}{r}

, then

m

equals:

Options

A $31$
B $32$
C $33$
D $34$

Correct answer

C. $33$

Step-by-step solution

Using the property

^{n}C_{r} = ^{n}C_{n-r}

, the given expression can be written as:

^{30}C_{r} + 3 \cdot ^{30}C_{r-1} + 3 \cdot ^{30}C_{r-2} + ^{30}C_{r-3}

This expression represents the coefficient of

x^r

in the expansion of:

(1+x)^{30} + 3x(1+x)^{30} + 3x^2(1+x)^{30} + x^3(1+x)^{30}

Taking

(1+x)^{30}

common, we get:

(1+x)^{30}(1 + 3x + 3x^2 + x^3)

= (1+x)^{30}(1+x)^3

= (1+x)^{33}

The coefficient of

x^r

in

(1+x)^{33}

is

^{33}C_{r}

.

Comparing this with

^{m}C_{r}

, we get

m = 33

.

Answer:

33

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