First slide

Binomial theorem for positive integral Index

Question

The value of $-^{15} C_{1} + {2.}^{15} C_{2} - {3.}^{15} C_{3} + ...... - {15.}^{15} C_{15} +^{14} C_{1} +^{14} C_{3} +^{14} C_{5} + ...... +^{14} C_{11}$ is

Moderate

A

$2^{13} - 14$
B

$2^{14}$
C

$2^{13} - 13$
D

$2^{16} - 1$

Solution

$\begin{matrix} S_{1} = -^{15} C_{1} + {2.}^{15} C_{2} - ...... - {15.}^{15} C_{15} = \sum_{r = 1}^{15} {(- 1)}^{r} . r .^{15} C_{r} = \sum_{r = 1}^{15} {(- 1)}^{r} . r . \frac{15}{r} {}^{14}C_{r} \\ = 15 (-^{14} C_{0} +^{14} C_{1} - .... -^{14} C_{14}) = 15 (0) = 0 \\ A l s o S_{2} =^{14} C_{1} +^{14} C_{3} + ...... +^{14} C_{11} \\ = (^{14} C_{1} +^{14} C_{3} + .... +^{14} C_{11} +^{14} C_{13}) -^{14} C_{13} \\ = 2^{13} - 14 \\ ∴ S_{1} + S_{2} = 2^{13} - 14 \end{matrix}$

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