Explanation

(c)
$f(x) = x^{n}\ \ \ \ \ \Rightarrow \ \ \ \ \ f(1) = 1$
$f^{'}(x) = nx^{n - 1}\ \ \ \ \ \ \ \Rightarrow \ \ \ f^{'}(1) = n$
$f^{''}(x) = n(n - 1)x^{n - 2}$ $\ \Rightarrow \ \ f^{''}(1) = n(n - 1)$
$\ \ \ldots\ \ldots\ \ldots\ \ldots\ \ldots\ \ \ldots\ \ldots$
$\ldots\ \ \ldots\ \ldots\ \ldots\ \ldots\ \ \ldots\ \ldots\$
$\ f^{n}(x) = n(n - 1)(n - 2)\ldots 2.1$
$\Rightarrow \ \ f^{n}(1) = n(n - 1)(n - 2)\ldots 2.1$
Now, $f(1) - \frac{f^{'}(1)}{1!} + \frac{f^{''}(1)}{2!} - \frac{f^{'''}(1)}{3!} + ... + \frac{( - 1)^{n}f^{n}(1)}{n!}$
$= 1 - \frac{n}{1!} + \frac{n(n - 1)}{2!} - \frac{n(n - 1)(n - 2)}{3!}\ + ... + \frac{( - 1)^{n}n(n - 1)(n - 2)\ldots 2.1}{n!}$
$= (1 - 1)^{n} = 0$