$\displaystyle\sum_{r=0}^{20}{ }^{50- r } C _{6}={ }^{50} C _{6}+{ }^{49} C _{6}+{ }^{48} C _{6}+\ldots . .+{ }^{30} C _{6}$
$={ }^{50} C _{6}+{ }^{49} C _{6}+\ldots . .+{ }^{31} C _{6}+\left({ }^{30} C _{6}+{ }^{30} C _{7}\right)-{ }^{30} C _{7}$
$={ }^{50} C _{6}+{ }^{49} C _{6}+\ldots . .+\left({ }^{31} C _{6}+{ }^{31} C _{7}\right)-{ }^{30} C _{7}$
$={ }^{50} C _{6}+{ }^{50} C _{7}-{ }^{30} C _{7}$
$={ }^{51} C _{7}-{ }^{30} C _{7}$
${ }^{ n } C _{ r }+{ }^{ n } C _{ r -1}={ }^{ n +1} C _{ r }$