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AP EAMCET Engineering 2018 Apr 23 Shift 2 Paper

Section: Mathematics

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Question : 17 of 160

Marks: +1, -0

If the coefficient of

x^{50}

in the expansion of

(1+x)^{100}+2x(1+x)^{99}+3x^{2}(1+x)^{98}

+ \dots + 101x^{100},

is

${}^{100}C_{50}$
${}^{101}C_{50}$
${}^{102}C_{50}$
${}^{103}C_{50}$

Solution:

Let,

S=(1+x)^{100}+2x(1+x)^{99}+3x^{2}(1+x)^{98}

+ \dots + 101x^{100}

\frac{x}{1+x} S = x(1+x)^{99}+2x^{2}(1+x)^{98}

+ \dots + 100x^{100}+\frac{101x^{101}}{1+x}

\frac{S}{1+x} = (1+x)^{100}+x(1+x)^{99}

+x^{2}(1+x)^{98}+ \dots + x^{100}-\frac{101x^{101}}{1+x}

\frac{S}{1+x} = \frac{(1+x)^{100} \left[ \left(\frac{x}{1+x}\right)^{101} - 1 \right]}{\frac{x}{1+x} - 1}

-\frac{101x^{101}}{1+x}

Simplify the above,

S = (1+x)^{102} - x^{102} - 102x^{101}

So, the coefficient of

x^{50} \text{ is } S = {}^{102}C_{50}

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