First slide

Binomial theorem for positive integral Index

Question

The coefficient of x⁴ in the expansion of ${(1 + x + x^{2} + x^{3})}^{n}$ is

Moderate

A

$^{n} C_{4}$
B

$^{n} C_{4} +^{n} C_{2}$
C

$^{n} C_{4} +^{n} C_{2} +^{n} C_{2}$
D

$^{n} C_{4} +^{n} C_{2} +^{n} C_{1} \cdot^{n} C_{2}$

Solution

${(1 + x + x^{2} + x^{3})}^{n} = \{(1 + x)^{n} {(1 + x^{2})}^{n}\} = (1 +^{n} C_{1} x +^{n} C_{2} x^{2} + \dots +^{n} C_{n} x^{n}) (1 +^{n} C_{1} x^{2} +^{n} C_{2} x^{4} + \dots +^{n} C_{n} x^{2 n})$

.Therefore, the coefficient of $\begin{aligned} x^{4} =^{n} C_{2} +^{n} {C_{2}}^{n} C_{1} +^{n} C_{4} \\ =^{n} C_{4} +^{n} C_{2} +^{n} {C_{1}}^{n} C_{2} \end{aligned}$

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