_ k =0 ^6 ^ 51- k C _3 is equal to - Vidyadip

MCQ

$\sum \limits_{ k =0}^6{ }^{51- k } C _3$ is equal to

A
${ }^{51} C _4-{ }^{45} C _4$
B
${ }^{51} C _3-{ }^{45} C _3$
✓
${ }^{52} C _4-{ }^{45} C _4$
D
${ }^{52} C _3-{ }^{45} C _3$

✓

Answer

Correct option: C.

${ }^{52} C _4-{ }^{45} C _4$

c
$\sum \limits_{ k =0}^6{ }^{51- k } C _3$

$={ }^{51} C _3+{ }^{50} C _3+{ }^{49} C _3+\ldots+{ }^{45} C _3$

$={ }^{45} C _3+{ }^{46} C _3+\ldots \ldots+{ }^{51} C _3$

$={ }^{45} C _4+{ }^{45} C _3+{ }^{46} C _3+\ldots \ldots+{ }^{51} C _3-{ }^{45} C _4$

$\left.={ }^n C _{ r }+{ }^n C _{ r -1}={ }^{ n +1} C _{ r }\right)$

$={ }^{52} C _4-{ }^{45} C _4$

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