First slide

Binomial theorem for positive integral Index

Question

The coefficient of t²⁴ in the expansion of ${(1 + t^{2})}^{12} (1 + t^{12}) (1 + t^{24})$ is

Moderate

A

$^{12} C_{6} + 2$
B

$^{12} C_{5}$
C

$^{12} C_{6}$
D

$^{12} C_{7}$

Solution

We have
$\begin{array}{l} {(1 + t^{2})}^{12} (1 + t^{12}) (1 + t^{24}) \\ = (1 +^{12} C_{1} t^{2} +^{12} C_{2} t^{4} + \dots +^{12} C_{6} t^{12} + \dots +^{12} C_{12} t^{24} + \dots) (1 + t^{12} + t^{24} + t^{36}) \end{array}$
$\begin{aligned} ∴ Coefficient of t^{24} in {(1 + t^{2})}^{12} (1 + t^{12}) (1 + t^{24}) \\ =^{12} C_{c} +^{12} C_{10} + 1 =^{12} C_{6} + 2 \end{aligned}$

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