JEE MAIN 2026 · Binomial Theorem and its Simple Applications

Turn each term into one coefficient. Using $\frac{1}{k+1}\binom{12}{k} = \frac{1}{13}\binom{13}{k+1}$, the general term is $\frac{2^{k+1}}{13}\binom{13}{k+1}$ for $k = 2, 4, \dots, 12$. So

$$26 \times (\dots) = 2\sum_{j \in \{3, 5, \dots, 13\}} \binom{13}{j} 2^{j}.$$

Odd-index sum. $\sum_{j \text{ odd}} \binom{13}{j}2^j = \frac{(1 + 2)^{13} - (1 - 2)^{13}}{2} = \frac{3^{13} + 1}{2}$. Remove the $j = 1$ term, $\binom{13}{1} \cdot 2 = 26$:

$$2\left(\frac{3^{13} + 1}{2} - 26\right) = 3^{13} + 1 - 52 = 3^{13} - 51.$$

Answer: $\alpha = 51$.