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JEE Main 202621 JanuaryEvening Shift
Exam
JEE Main
Year
2026
Date
21 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2026 · 21 January · Evening Shift · Mathematics · Binomial Theorem

If ( 1 ^ 15 C_ 0 + 1 ^ 15 C_ 1 )( 1 ^ 15 C_ 1 + 1 ^ 15 C_ 2 ) ( 1 ^...

If (115C0+115C1)(115C1+115C2)⋯(115C12+115C13)=α1314C014C1⋯14C12\left(\frac{1}{{ }^{15} \mathrm{C}_{0}}+\frac{1}{{ }^{15} \mathrm{C}_{1}}\right)\left(\frac{1}{{ }^{15} \mathrm{C}_{1}}+\frac{1}{{ }^{15} \mathrm{C}_{2}}\right) \cdots\left(\frac{1}{{ }^{15} \mathrm{C}_{12}}+\frac{1}{{ }^{15} \mathrm{C}_{13}}\right)=\frac{\alpha^{13}}{{ }^{14} \mathrm{C}_{0}{ }^{14} \mathrm{C}_{1} \cdots{ }^{14} \mathrm{C}_{12}}, then 30α30 \alpha is equal to ____\_\_\_\_.

Correct answer

0

Step-by-step solution

The general term of the product is Tr=115Cr−1+115CrT_r = \frac{1}{^{15}C_{r-1}} + \frac{1}{^{15}C_r} for r=1,2,…,13r = 1, 2, \dots, 13.
Using the formula nCr=nr⋅n−1Cr−1^{n}C_r = \frac{n}{r} \cdot ^{n-1}C_{r-1}, we have 15Cr=15r⋅14Cr−1^{15}C_r = \frac{15}{r} \cdot ^{14}C_{r-1} and 15Cr−1=1515−(r−1)⋅14Cr−1=1516−r⋅14Cr−1^{15}C_{r-1} = \frac{15}{15-(r-1)} \cdot ^{14}C_{r-1} = \frac{15}{16-r} \cdot ^{14}C_{r-1}.
Substituting these into TrT_r:
Tr=16−r15⋅14Cr−1+r15⋅14Cr−1=16−r+r15⋅14Cr−1=1615⋅14Cr−1T_r = \frac{16-r}{15 \cdot ^{14}C_{r-1}} + \frac{r}{15 \cdot ^{14}C_{r-1}} = \frac{16-r+r}{15 \cdot ^{14}C_{r-1}} = \frac{16}{15 \cdot ^{14}C_{r-1}}.
The product is ∏r=113Tr=∏r=1131615⋅14Cr−1=16131513⋅∏r=11314Cr−1\prod_{r=1}^{13} T_r= \prod_{r=1}^{13} \frac{16}{15\cdot {^{14}C_{r-1}}}= \frac{16^{13}}{15^{13} \cdot \prod_{r=1}^{13}{^{14}C_{r-1}}}.
The denominator of the product is 1513⋅(14C0⋅14C1…14C12)15^{13} \cdot (^{14}C_0 \cdot ^{14}C_1 \dots ^{14}C_{12}).
Comparing this with the given expression α1314C0⋅14C1…14C12\frac{\alpha^{13}}{^{14}C_0 \cdot ^{14}C_1 \dots ^{14}C_{12}}, we get:
α13=16131513  ⟹  α=1615\alpha^{13} = \frac{16^{13}}{15^{13}} \implies \alpha = \frac{16}{15}.
We need to find the value of 30α30\alpha:
30α=30×1615=2×16=3230\alpha = 30 \times \frac{16}{15} = 2 \times 16 = 32.
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