First slide

Binomial theorem for positive integral Index

Question

If n € N, n > 1, then value of
$\begin{array}{l} E = a -^{n} C_{1} (a - 1) +^{n} C_{2} (a - 2) + \dots . . . . + (- 1)^{n} {(a - n)}^{n} C_{n} is \end{array}$

Moderate

A

$α$
B

0
C

$α^{2}$
D

2ⁿ

Solution

Given equation can be rewritten as
$E = a [^{n} C_{0} -^{n} C_{1} +^{n} C_{2} - \dots + (- 1)^{n n} C_{n}]$ $+ [^{n} C_{1} - (2) (^{n} C_{2}) + (3) (^{n} C_{3}) - \dots + (- 1)^{n} (n) (^{n} C_{n})]$
$\Rightarrow E = 0 + 0 = 0$ $[by properties]$

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