• # question_answer If $x,\ y$ and $r$ are positive integers, then $^{x}{{C}_{r}}{{+}^{x}}{{C}_{r-1}}^{y}{{C}_{1}}{{+}^{x}}{{C}_{r-2}}^{y}{{C}_{2}}+.......{{+}^{y}}{{C}_{r}}=$ [Karnataka CET 1993; RPET 2001] A) $\frac{x\ !\ y\ !}{r\ !}$ B) $\frac{(x+y)\ !}{r\ !}$ C) $^{x+y}{{C}_{r}}$ D) $^{xy}{{C}_{r}}$

The result is trivially true for$r=1,\ 2$. It can be easily proved by the principle of mathematical induction that the result is true for $r$ also.