If ( 1 C 0 15 + 1 C 1 15 ) ( 1 C 1 15 + 1 C 2 15 ) ? ( 1 C 12 15 + 1 C 13 15 ) = C ? a 13 C 0 14 ? 14 C 1 ? ? 14 C 12 , then 30 is equal to_____ [2026]

Question

If $((\frac{1}{{}^{15}{C_{0}}} + \frac{1}{{}^{15}{C_{1}}})) ((\frac{1}{{}^{15}{C_{1}}} + \frac{1}{{}^{15}{C_{2}}})) \dots ((\frac{1}{{}^{15}{C_{12}}} + \frac{1}{{}^{15}{C_{13}}})) = \frac{α^{13}}{{}^{14}{C_{0}}^{14} C_{1} \dots^{14} C_{12}},$ then $30 α$ is equal to_____ [2026]

Smart Achievers · Accepted Answer

(32)

$\prod_{r = 0}^{12} (\frac{1}{{}^{15}{C_{r}}} + \frac{1}{{}^{15}{C_{r + 1}}}) = \prod_{r = 0}^{12} \frac{\frac{16}{r + 1} \cdot {}^{15}C_{r}}{15 C_{r} \cdot {}^{15}C_{r + 1}}$

$= \prod_{r = 0}^{12} \frac{16}{(r + 1) \cdot \frac{15}{r + 1} \cdot {}^{14}C_{r}} = \prod_{r = 0}^{12} \frac{\frac{16}{15}}{{}^{14}C_{r}}$

$= \frac{{(\frac{16}{15})}^{13}}{{}^{14}C_{0} \cdot {}^{14}C_{1} \dots {}^{14}C_{12}}$ $\Rightarrow α = \frac{16}{15}$

$\Rightarrow 30 α = 32$

Solution

Q.
If $((\frac{1}{{}^{15}{C_{0}}} + \frac{1}{{}^{15}{C_{1}}})) ((\frac{1}{{}^{15}{C_{1}}} + \frac{1}{{}^{15}{C_{2}}})) \dots ((\frac{1}{{}^{15}{C_{12}}} + \frac{1}{{}^{15}{C_{13}}})) = \frac{α^{13}}{{}^{14}{C_{0}}^{14} C_{1} \dots^{14} C_{12}},$ then $30 α$ is equal to_____ [2026]

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Solution

Q. If (1C015+1C115)(1C115+1C215)⋯(1C1215+1C1315)=α13C014 14C1⋯ 14C12, then 30α is equal to_____ [2026]

Ans. (32) ∏r=012(1Cr15+1Cr+115)=∏r=01216r+1·Cr1515Cr·Cr+115 =∏r=01216(r+1)·15r+1·Cr14=∏r=0121615Cr14 =(1615)13C014·C114⋯C1214⇒α=1615 ⇒30α=32

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Q.
If $((\frac{1}{{}^{15}{C_{0}}} + \frac{1}{{}^{15}{C_{1}}})) ((\frac{1}{{}^{15}{C_{1}}} + \frac{1}{{}^{15}{C_{2}}})) \dots ((\frac{1}{{}^{15}{C_{12}}} + \frac{1}{{}^{15}{C_{13}}})) = \frac{α^{13}}{{}^{14}{C_{0}}^{14} C_{1} \dots^{14} C_{12}},$ then $30 α$ is equal to_____ [2026]