The coefficient of x60 in (1+x)511−x+x250 is

The coefficient of $x^{60} in (1 + x)^{51} {(1 - x + x^{2})}^{50}$ is

A
$^{50} C_{20}$
B
$- (^{50} C_{20})$
C
$^{51} C_{20}$
D
$- (^{51} C_{20})$

Solution:

The coefficient of $x^{60}$ in $(1 + x)^{51} {(1 - x + x^{2})}^{50}$

= the coefficient of $x^{60}$ in $(1 + x) {(1 - x^{3})}^{50}$

$=^{50} C_{20}$

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