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Q.
$\displaystyle\sum_{k=0}^6{ }^{51-k} C_3$ is equal to
JEE MainJEE Main 2023Permutations and Combinations
Solution:
$ \displaystyle\sum_{ k =0}^6{ }^{51- k } C _3$
$ ={ }^{51} C _3+{ }^{50} C _3+{ }^{49} C _3+\ldots+{ }^{45} C _3$
$ ={ }^{45} C _3+{ }^{46} C _3+\ldots . .+{ }^{51} C _3 $
$ ={ }^{45} C _4+{ }^{45} C _3+{ }^{46} C _3+\ldots . .+{ }^{51} C _3-{ }^{45} C _4 $
$ \left({ }^n C_r+{ }^n C_{r-1}={ }^{n+1} C_r\right)$
$ ={ }^{52} C _4-{ }^{45} C _4 $