Find sum of series $^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . . . - . . . . . +^{20} C_{10}$

Open in App

Solution

To find:- $^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . . . - . . . . . +^{20} C_{10}$
Solution:-
As we know that,
${(1 + x)}^{20} =^{20} C_{0} +^{20} C_{1} x +^{20} C_{2} x^{2} +^{20} C_{3} x^{3} + . . . . . +^{20} C_{20} x^{20}$
Substituting $x = - 1$ in the above equation, we have
${(1 + (- 1))}^{20} =^{20} C_{0} +^{20} C_{1} (- 1) +^{20} C_{2} {(- 1)}^{2} + . . . . . +^{20} C_{20} {(- 1)}^{20}$
$0 =^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . . . - . . . . . +^{20} C_{20}$
$\Rightarrow 2 (^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . . . -^{20} C_{9}) +^{20} C_{10} = 0$
$\Rightarrow 2 (^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . . . -^{20} C_{9} +^{20} C_{10}) =^{20} C_{10}$
$\Rightarrow^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . . . -^{20} C_{9} +^{20} C_{10} = \frac{1}{2}^{20} C_{10}$
Hence the value of $^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . . . - . . . . . +^{20} C_{10}$ is $\frac{1}{2}^{20} C_{10}$ .

Suggest Corrections

Similar questions

{^{20} C}_{0} - {^{20} C}_{1} + {^{20} C}_{2} - {^{20} C}_{3} + . . . . + {^{20} C}_{10}

^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . . . - . . . . . . +^{20} C_{10}

^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + . . . +^{20} C_{10} =

The sum of the series $^{20} C_{0} +^{20} C_{1} +^{20} C_{2} + . . . +^{20} C_{10}$ is

^{20} C_{0} -^{20} C_{1} -^{20} C_{2} -^{20} C_{3} . . . . . . . .^{20} C_{10} i s

Join BYJU'S Learning Program