The coefficient of $x^{10}$ in
$(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}$ is

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Solution

The correct option is C 18
The following expression is Geometric progression with common ratio as $x$
Therefore $(x + x^{2} + x^{3} + x^{4} + x^{5})^{3}$
$= (x (1 + x + x^{2} + x^{3} + x^{4}))^{3}$
$= (x (\frac{1 - x^{5}}{1 - x}))^{3}$
$= (\frac{x - x^{6}}{1 - x})^{3}$
$= (x - x^{6})^{3} (1 - x)^{- 3}$
$= (x^{3} - x^{18} - 3 x^{8} + 3 x^{13}) (1 + 3 x + 6 x^{2} + 10 x^{3} . . . + 36 x^{7} . . . \infty)$
Hence coefficient of $x^{10}$ will be
sum of coefficient of $x^{3} . x^{7}$ and $- x^{8} . x^{2}$
$= 36 - 3 (6)$
$= 18$

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The coefficient of x10 in (x+x2+x3+x4+x5)3 is

The coefficient of $x^{10}$ in
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