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JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Binomial Theorem

The sum of all rational terms in the expansion of

(2 + 3)^{8}

is

(A)16923
(B)3763
(C)33845
(D)18817✓ Correct

Answer: (D)

Solution

In

(2 + 3)^{8} = \sum_{r = 0}^{8} (r 8) 2^{8 - r} (3)^{r}

, a term is rational exactly when

r

is even. Summing over

r = 0, 2, 4, 6, 8

,

(0 8) 2^{8} + (2 8) 2^{6} \cdot 3 + (4 8) 2^{4} \cdot 9 + (6 8) 2^{2} \cdot 27 + (8 8) 81 = 256 + 5376 + 10080 + 3024 + 81.

The sum of all rational terms is

18817

.

JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Single correct option

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