Binomial Theorem
Product of Binomial Ratios — Finding 30α
nta_pyq_2026_jan
Grade 11
Question:
If $\left(\dfrac{1}{{}^{15}C_0}+\dfrac{1}{{}^{15}C_1}\right)\left(\dfrac{1}{{}^{15}C_1}+\dfrac{1}{{}^{15}C_2}\right)\cdots\left(\dfrac{1}{{}^{15}C_{12}}+\dfrac{1}{{}^{15}C_{13}}\right)=\dfrac{\alpha^{13}}{{}^{14}C_0\cdot{}^{14}C_1\cdots{}^{14}C_{12}}$, then $30\alpha$ is equal to _____
Step-by-Step Solution
Key Concept: General term: $T_r=\frac{1}{{}^{15}C_{r-1}}+\frac{1}{{}^{15}C_r}$. Using ${}^{15}C_r=\frac{15}{r}\cdot{}^{14}C_{r-1}$: $T_r=\frac{16}{15\cdot{}^{14}C_{r-1}}$. Product $=\prod_{r=1}^{13}T_r=\frac{16^{13}}{15^{13}\cdot\prod_{r=1}^{13}{}^{14}C_{r-1}}$.
$\alpha=16/15$. $30\alpha=32$.
Correct Answer: 32