First slide

Binomial theorem for positive integral Index

Question

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

Moderate

A

$^{12} C_{6} + 13$
B

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

$^{12} C_{6} + 1$
D

$^{12} C_{6}$

Solution

Coefficient to $t^{24} in {(1 + t^{2})}^{12} (1 + t^{12}) (1 + t^{24})$
Coefficient of $t^{24} in (1 +^{12} C_{1} t^{2} +^{12} C_{2} t^{4} + \dots)$ $(1 + t^{12} + t^{24})$
$=^{12} C_{12} +^{12} C_{6} + 1 =^{12} C_{6} + 2$

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