The coefficient of $t^{32}$ in the expansion of ${(1 + t^{2})}^{12} (1 + t^{12}) (1 + t^{24})$ is

^{12} C_{10} +^{12} C_{4}

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^{12} C_{5}

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^{12} C_{6}

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^{12} C_{7}

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Solution

The correct option is A $^{12} C_{10} +^{12} C_{4}$
${(1 + t^{2})}^{12} (1 + t^{12}) (1 + t^{24})$
$= (1 + t^{12}) (1 + t^{24}) {(1 + t^{2})}^{12}$
$= (1 + t^{12} + t^{24} + t^{36})$ expansion of ${(1 + t^{2})}^{12}$
$= (1 + t^{12} + t^{24} + t^{36}) (1 + 12_{C_{1}} t^{2} + 12_{C_{2}} {(t^{2})}^{2} + 12_{C_{3}} {(t^{2})}^{3} + 12_{C_{4}} {(t^{2})}^{4} + 12_{C_{5}} {(t^{2})}^{5} + 12_{C_{6}} {(t^{2})}^{6} + 12_{C_{7}} {(t^{2})}^{7} + 12_{C_{8}} {(t^{2})}^{8} + 12_{C_{9}} {(t^{2})}^{9} + 12_{C_{10}} {(t^{2})}^{10} + 12_{C_{11}} + 12_{C_{12}} {(t^{2})}^{12})$
$= (1 + t^{12} + t^{24} + t^{36}) (1 + 12_{C_{1}} t^{2} + 12_{C_{2}} t^{4} + 12_{C_{3}} t^{6} + 12_{C_{4}} t^{8} + 12_{C_{5}} t^{10} + 12_{C_{6}} t^{10} + 12_{C_{7}} t^{14} + 12_{C_{8}} t^{16} + 12_{C_{9}} t^{18} + 12_{C_{10}} t^{20} + 12_{C_{11}} t^{22} + 12_{C_{12}} t^{24})$
Terms containing $t^{32}$ is
$= (t^{24}) 12_{C_{4}} t^{8} + (t^{12}) 12_{C_{10}} t^{20}$
Co-efficients $^{12} C_{10} +^{12} C_{4}$

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Similar questions

t^{32}

{(1 + t^{2})}^{12} (1 + t^{12}) (1 + t^{24})

t^{24}

(1 + t^{2})^{12} (1 + t^{12}) (1 + t^{24})

t^{24}

(1 + t^{2})^{12} (1 + t^{12}) (1 + t^{24})

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{(1 + t^{2})}^{12} (1 + t^{12}) (1 + t^{24})

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