Binomial theorem for positive integral Index

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Coefficient of $x^{4}$ in ${(1 + x + x^{2} + x^{3})}^{11}$ is-

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Solution

We have
$\begin{array}{l} 1 + x + x^{2} + x^{3} = 1 + x + x^{2} (1 + x) = (1 + x) (1 + x^{2}) \\ \Rightarrow {(1 + x + x^{2} + x^{3})}^{11} = (1 + x)^{11} {(1 + x^{2})}^{11} \\ = (1 +^{11} C_{1} x +^{11} C_{2} x^{2} +^{11} C_{3} x^{3} + \\ ^{11} C_{4} x^{4} + \dots) (1 +^{11} C_{1} x^{2} + \\ ^{11} C_{2} x^{4} + \dots) \end{array}$
Thus, coefficient of $x^{4}$ in ${(1 + x + x^{2} + x^{3})}^{11}$
$\begin{array}{l} = (1) (^{11} C_{2}) + (^{11} C_{2}) (^{11} C_{1}) + (^{11} C_{4}) (1) \\ = 55 + (55) (11) + 330 = 990 \end{array}$

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Binomial theorem for positive integral Index

Remember concepts with our Masterclasses.

Coefficient of x4 in 1+x+x2+x311is-

We have 1+x+x2+x3=1+x+x2(1+x)=(1+x)1+x2⇒1+x+x2+x311=(1+x)111+x211=1+11C1x+11C2x2+11C3x3+ 11C4x4+⋯1+11C1x2+ 11C2x4+⋯Thus, coefficient of x4 in 1+x+x2+x311=(1) 11C2+ 11C2 11C1+ 11C4(1)=55+(55)(11)+330=990

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Coefficient of $x^{4}$ in ${(1 + x + x^{2} + x^{3})}^{11}$ is-