Find the coefficient of $x^{10}$ in $(1 - x^{7}) (1 - x^{8}) (1 - x^{9}) (1 - x)^{- 3}$

^{12} C_{10}

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12 C_{2} -^{5} C_{3} -^{4} C_{2} -^{3} C_{1}

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^{12} C_{2}

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^{12} C_{2} -^{5} C_{2} -^{4} C_{2} -^{3} C_{3}

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Solution

The correct option is D $^{12} C_{2} -^{5} C_{2} -^{4} C_{2} -^{3} C_{3}$
Coefficient of $x^{10}$ in $(1 - x^{7}) (1 - x^{8}) (1 - x^{9}) (1 - x)^{- 3}$

= $(1 - x^{7}) (1 - x^{8}) (1 - x^{9})$ $(1 +^{3} C_{1} x +^{4} C_{2} x^{2} + . . . . . . . . +^{12} C_{10} x^{10} + . . . . . . . . . .)$

Ignoring the powers higher than $x^{10}$

= $(1 - x^{7} - x^{8} + x^{15}) (1 - x^{9})$ $(1 +^{3} C_{1} x +^{4} C_{2} x^{2} + . . . . . . . . . . . +^{12} C_{10} x^{10} + . . . . . . . . . . . . .)$

= $(1 - x^{7} - x^{8}) (1 - x^{9})$ $(1 +^{3} C_{1} x +^{4} C_{2} x^{2} + . . . . . . . . . . +^{12} C_{10} x^{10} + . . . . . . . . . . . . . .)$

Coefficient of $x^{10}$ in $(1 - x^{7} - x^{8} - x^{9})$ $(1 +^{3} C_{1} x +^{4} C_{2} x^{2} + . . . . . . . . . . . . +^{12} C_{10} x^{10} + . . . . . . . . . . . . . . . .)$

$^{12} C_{2} -^{5} C_{3} -^{4} C_{2} -^{3} C_{1} =^{12} C_{2} -^{5} C_{2} -^{4} C_{2} -^{3} C_{2}$

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Similar questions

One or more options can be correct:
Find the coefficient of $x^{10}$ in $(1 - x^{7}) (1 - x^{8}) (1 - x^{9}) (1 - x)^{- 3}$

Find the coefficient of $x^{8} i n (1 + x)^{8} (1 - x)^{- 2}$ .

∣ ∣ ∣ ∣ ∣ \begin{matrix} (1 + x) & (1 + x)^{2} & (1 + x)^{3} (1 + x)^{4} & (1 + x)^{5} & (1 + x)^{6} (1 + x)^{7} & (1 + x)^{8} & (1 + x)^{9} \end{matrix} ∣ ∣ ∣ ∣ ∣

= a_{0} + a_{1} x + a_{2} x^{2} + \dots \dots

a_{1}

x^{10}

(1 + x)^{- 1}

x^{10}

(1 - x^{2})^{10}

x

{(x - \frac{2}{x})}^{10}

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