Binomial theorem for positive integral Index

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Question

The coefficient of $x^{5}$ in the expansion of ${(1 + x^{2})}^{5} (1 + x)^{4}$ is

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Solution

${(1 + x^{2})}^{5} (1 + x)^{4}$
$\begin{array}{l} = (^{5} C_{0} +^{5} C_{1} x^{2} +^{5} C_{2} x^{4} + \dots) \\ x (^{4} C_{0} +^{4} C_{1} x +^{4} C_{2} x^{2} +^{4} C_{3} x^{3} +^{4} C_{4} x^{4}) \end{array}$
$∴$ Coefficient of $x^{5}$ in $\{{(1 + x^{2})}^{5} (1 + x)^{4}\}$
$=^{5} C_{2} \times^{4} C_{1} +^{4} C_{3} \times^{5} C_{1} = 60$

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Binomial theorem for positive integral Index

Remember concepts with our Masterclasses.

The coefficient of x5 in the expansion of 1+x25(1+x)4 is

1+x25(1+x)4= 5C0+5C1x2+5C2x4+…x 4C0+4C1x+4C2x2+4C3x3+4C4x4∴ Coefficient of x5 in 1+x25(1+x)4=5C2×4C1+4C3×5C1=60

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The coefficient of $x^{5}$ in the expansion of ${(1 + x^{2})}^{5} (1 + x)^{4}$ is