If $f (x) = 1 + x^{2} - x^{3} + . . . - x^{15} + x^{16} - x^{17}$ , then the coefficient of $x^{2}$ in $f (x - 1)$ is

826

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816

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822

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none of these

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Solution

The correct option is B $816$
The coefficient of $x^{2}$ will be
$^{2} C_{2} +^{3} C_{2} +^{4} C_{2} +^{5} C_{2} + . . .^{17} C_{2}$
Applying $^{n} C_{r} +^{n} C_{r + 1} =^{n + 1} C_{r + 1}$
The above expression simplifies to
$[1 +^{3} C_{3} +^{3} C_{2} +^{4} C_{2} +^{5} C_{2} + . . .^{17} C_{2}] -^{3} C_{3}$
$= [1 +^{17} C_{3} +^{17} C_{2}] - 1$
$= 1 +^{18} C_{3} - 1$
$=^{18} C_{3}$
$= 816$

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Similar questions

f (x) = 1 - x + x^{2} - x^{3} + . . . . - x^{15} + x^{16} + x^{17}

x^{2}

f (x - 1)

f (x) = 1 - x + x^{2} - x^{3} + \dots \dots - x^{15} + x^{16} - x^{17}

x^{2}

f (x - 1)

f (x) = 1 - x + x^{2} - x^{3} + \dots - x^{15} + x^{16} - x^{17},

x^{2}

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1 - x + x^{2} - x^{3} + . . . . . . . + x^{16} - x^{17},

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1 + x^{2} + x^{4} + x^{6} + \dots + x^{34}

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