Question

In the expansion of $(1+x{)}^n$, what is the sum of even binomial coefficients ?

• A. $2^n$
• B. $2^{n-1}$
• C. $2^{n+1}$
• D. None of the above

Answer 1

The correct option is B $2^{n-1}$

Now,

$(1+x{)}^n={}^n{C}_0+x^1\times {}^n{C}_1+x^2\times {}^n{C}_2.........+x^n\times {}^n{C}_n$

Putting $x=-1$

$(0{)}^n={}^n{C}_0-{}^n{C}_1+{}^n{C}_2.........+{(-1)}^n\times {}^n{C}_n$

Putting $x=1$

$(2{)}^n={}^n{C}_0+{}^n{C}_1+{}^n{C}_2.........+{}^n{C}_n$

Adding both we have,

$(2{)}^n=2\times {}^n{C}_0+2\times {}^n{C}_2.........$

$(2{)}^{n-1}={}^n{C}_0+{}^n{C}_2.........$

$=$ sum of even binomial coefficients.