If n is an integer greater than 1, show that
$a -^{n} C_{1} (a - 1) +^{n} C_{2} (a - 2) - . . . . . + (- 1)^{n} (a - n) = 0$

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Solution

$a - C_{1} (a - 1) + C_{2} (a - 2) + C_{3} (a - 3) + . . . . . . . . . . + (- 1)^{n} (a - n) = 0$
Now $C_{0} = 1, C_{n} = 1$ Collect terms of a in one bracket. and without a in the other.
$L . H . S = a [C_{0} - C_{1} + C_{2} - C_{3} + . . . . . . . (- 1)^{n} . C_{n}] + [C_{1} - 2 C_{2} + 3 C_{3} + . . . . . . . (- 1)^{n - 1} n C_{n}]$
$∵ - (- 1)^{n} = (- 1) (- 1) (- 1)^{n - 1} = (- 1)^{n - 1}$
$L . H . S = a 0 + 0 = 0$

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