Coefficient of x 99 in x 100 + 2 x 99 ( 1 + x ) + 3 ( 1 + x ) 2 x 98 + ? + 101 ( 1 + x ) 100 is

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Solution

Q.
$Coefficient of x^{99} in x^{100} + 2 x^{99} (1 + x) + 3 {(1 + x)}^{2} x^{98} + \dots + 101 {(1 + x)}^{100} is____$

1 ${}^{101}C_{99} - {}^{101}C_{98}$

2 $100 ({}^{101}C_{99}) - {}^{101}C_{98}$

3 ${}^{- 101}C_{98} + 101 ({}^{101}C_{99})$

4 $101 ({}^{101}C_{98}) - {}^{101}C_{99}$

Ans.
(3)

$Required = coefficient of x^{99} in x^{100} (1 + 2 (\frac{1 + x}{x}) + . . . . . 101 {(\frac{1 + x}{x})}^{100})$

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