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Binomial Expression

The coeffcien...

Question

The coeffcient of $x^{10}$ in the expansion of ${(1 + x)}^{2} {(1 + x^{2})}^{3} {(1 + x^{3})}^{4}$ is equal to

52

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Solution

The correct option is A $52$
To find the co-efficient of $x^{10}$ in express of ${1 + x}^{2} (1 + x^{2})^{3} (1 + x^{3})^{4}$
The above expression can be expanded as
$(1 + 2 x + x^{2}) (1 + 3 x^{2} + 3 x^{4} + x^{6}) (1 + 4 x^{3} + 6 x^{6} + 4 x^{3} + x^{12})$
The $x^{10}$ common when
$= 3 x^{4} + 6 x^{6} + 2 x \times 4 x^{3} + x^{2} \times 3 x^{2} \times 6 x^{6} + 2 x . x^{6} 4 x^{3}$
$= 18 x^{10} + 8 x^{10} + 18 x^{10} + 8 x^{10}$
$= 52 x^{10}$
Hence the answer is $52$

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