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  4. The value of 14C1 + 14C3 + 14C5 + ..... + 14C11 is

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Q. The value of $^{14}C_1 + \,{}^{14}C_3 + \,{}^{14}C_5 + ..... + \,{}^{14}C_{11}$ is

Binomial Theorem

Solution:

$^{14}C_{1}+\,{}^{14}C_{3}+\,{}^{14}C_{5}+....+\,{}^{14}C_{11}$
$= \left(\,{}^{14}C_{1}+\,{}^{14}C_{3}+\,{}^{14}C_{5}+.... +\,{}^{14}C_{13}\right)-\,{}^{14}C_{13}$
$= 2^{14-1}-\,{}^{14}C_{1} = 2^{13}-14$.