Question

The sum of the coefficients of all the terms in the expansion of $(2x-y+z{)}^{20}$, in which the power of $y$ is 0 and $x$ appears in even powers and $z$ appears in odd powers, is

• A. $0$
• B. $\frac{2^{20}-1}{2}$
• C. $2^{19}$
• D. $\frac{3^{20}-1}{2}$

Answer 1

Answer: (A)

$0$

The general term in the expansion of $(2x-y+z{)}^{20}$ is $\left(2x{)}^p\right(-y{)}^q(z{)}^r$.
$p+q+r=20$
$q=0$
$p+r=20$
According to the given condition, $p$ should be even and $r$ should be odd. The sum of even and odd is odd which is not possible.
Hence, no such term exists.
Hence, the sum of coefficients is zero.