First slide

Binomial theorem for positive integral Index

Question

$The sum of the coefficient of all the terms in the expansion$ $of (2 x - y + z)^{20} in which y do not appear at all while x$ $appears in even powers and z appears in odd powers is -$

Moderate

A

0
B

$\frac{2^{20} - 1}{2}$
C

$2^{19}$
D

$\frac{3^{20} - 1}{2}$

Solution

$\begin{array}{l} c o e f f i c i e n t o f g e n e r a l t e r m = \frac{20!}{p! q! r!} = \frac{20!}{p! q! r!} 2^{p} x^{p} y^{q} z^{r} \\ where p + q + r = 20, q = 0 \\ ⇒p + r = 20 (p is even and r is odd). \\ ⇒even + odd = even (not possible) \\ ⇒ Coefficient of such power never occur \\ ∴ coefficient is zero \end{array}$

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