Question
The value of $$^{20}{C_0} + {\,^{20}}{C_1} + {\,^{20}}{C_2} + {\,^{20}}{C_3} + {\,^{20}}{C_4} + {\,^{20}}{C_{12}} + {\,^{20}}{C_{13}} + {\,^{20}}{C_{14}} + {\,^{20}}{C_{15}}$$ is
A.
$${2^{19}} - \frac{{\left( {^{20}{C_{10}} + {\,^{20}}{C_9}} \right)}}{2}$$
B.
$${2^{19}} - \frac{{\left( {^{20}{C_{10}} + 2 \times {\,^{20}}{C_9}} \right)}}{2}$$
C.
$${2^{19}} - \frac{{^{20}{C_{10}}}}{2}$$
D.
None of these
Answer :
$${2^{19}} - \frac{{\left( {^{20}{C_{10}} + 2 \times {\,^{20}}{C_9}} \right)}}{2}$$
Solution :
Given series is $$^{20}{C_0} + {\,^{20}}{C_1} + {\,^{20}}{C_2} + ..... + {\,^{28}}{C_8}$$
$$\eqalign{
& = \frac{{\left( {{2^{20}} - {\,^{20}}{C_{10}}} \right)}}{2} - {\,^{20}}{C_9} \cr
& = {2^{19}} - \frac{{\left( {^{20}{C_{10}} + 2 \times {\,^{20}}{C_9}} \right)}}{2} \cr} $$