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6.Permutation and Combination
hard
The value of $\sum \limits_{ r =0}^{20}{ }^{50- r } C _{6}$ is equal to
A
${ }^{51} C _{7}+{ }^{30} C _{7}$
B
${ }^{51} C _{7}-{ }^{30} C _{7}$
C
${ }^{50} C _{7}-{ }^{30} C _{7}$
D
$^{50} C _{6}-{ }^{30} C _{6}$
(JEE MAIN-2020)
Solution
$\sum_{r=0}^{20}{ }^{50-r} C_{6}={ }^{50} C_{6}+{ }^{49} C_{6}+{ }^{48} C_{6}+\ldots . .+{ }^{30} C_{6}$
$={ }^{50} C_{6}+{ }^{49} C_{6}+\ldots . .+{ }^{31} C_{6}+\left({ }^{30} C_{6}+{ }^{30} C_{7}\right)-{ }^{30} C_{7}$
$={ }^{50} C_{6}+{ }^{49} C_{6}+\ldots . .+\left({ }^{31} C_{6}+{ }^{31} C_{7}\right)-{ }^{30} C_{7}$
$={ }^{50} C_{6}+{ }^{50} C_{7}-{ }^{30} C_{7}$
$={ }^{51} C_{7}-{ }^{30} C_{7}$
${ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}$
Standard 11
Mathematics
