First slide

Binomial theorem for positive integral Index

Question

Coefficient of $x^{17}$ in the polynomial $P (x)$ $= \prod_{r = 0}^{17} (x +^{35} C_{r})$ is

Moderate

A

$2^{34}$
B

$^{36} C_{17}$
C

$2^{35} -^{36} C_{17}$
D

0

Solution

$P (x)$ is a polynomial of degree $18$ , and coefficient of $x^{17}$ is
$N =^{35} C_{0} +^{35} C_{1} +^{35} C_{2} + \dots +^{35} C_{17} (1)$
Using $^{n} C_{r} =^{n} C_{n - r}$ , we can write $N$ as
$N =^{35} C_{35} +^{35} C_{34} + \dots +^{35} C_{18} (2)$
Adding $(1)$ and $(2)$ , we get
$\begin{array}{l} 2 N =^{35} C_{0} +^{35} C_{1} + \dots +^{35} C_{35} = 2^{35} \\ \Rightarrow N = 2^{34} \end{array}$

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