If $a={99}^{50}+{100}^{50}$ and $b={101}^{50}$ then 

$\begin{array}{l}{101}^{50}-{99}^{50}=(100+1{)}^{50}-(100-1{)}^{50}\\ =2\left[{ }^{50}{C}_1\left(100{)}^{49}{+}^{50}{C}_3\right(100{)}^{47}+\dots {+}^{50}{C}_{49}\left(100\right)\right]\end{array}$\

$={100}^{50}+$ a positive term.

$\Rightarrow b = a +$(a positive term) $\Rightarrow a < b.$