we have for y= $(x-1)^n$

$y_1 $ = $n(x-1)^{n-1}$

$y_2$ = $n(n-1)(x-1)^{n-2}$

:

:

$y_n = n(n-1)(n-2)!....(n-(n-1))(x-1)^{n-n} = n!$

hence  $ y + \frac{y_1}{1!} + \frac{y_2}{2!} + \frac{y_3}{3!} + ....+ \frac{y_n}{n!}$

$=(x-1)^n + \frac{n}{1!}(x-1)^{n-1} + \frac{n(n-1)}{2!}(x-1)^{n-2} + \frac{n(n-1)(n-2)}{3!}(x-1)^{n-3} +.....+ \frac{n!}{n!}$

$ = (x-1)^n + ^nC_1(x-1)^{n-1}.1 + ^nC_2(x-1)^{n-2}.{(1)}^2 + ^nC_3(x-1)^{n-3}.{(1)}^3+....+{(1)^n}$

= ${[(x-1) + 1]}^n$

= $x^n $

Hence, proved