JEE Main 2026 — Apr 2, Shift 2 — Mathematics solution

JEE Main2026Apr 2, Shift 2Mathematics

If for $3 \leq r \leq 30$ , $(^{30} C_{30 - r}) + 3 (^{30} C_{31 - r}) + 3 (^{30} C_{32 - r}) + (^{30} C_{33 - r}) =^{m} C_{r}$ , then $m$ equals :

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

Solution

Recognise the coefficients $1, 3, 3, 1$ as binomial coefficients $^{3} C_{0},^{3} C_{1},^{3} C_{2},^{3} C_{3}$ . The expression becomes the Vandermonde-style sum

$\sum_{k = 0}^{3}^{3} C_{k} \cdot^{30} C_{30 - r + k} =^{33} C_{33 - r}$ .

Since $^{33} C_{33 - r} =^{33} C_{r}$ , comparing with $^{m} C_{r}$ gives $m = 33$ .

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