Coefficient of t^{20} in expansion of (1 + t^2)^{10}(1 + t^{10})(1 + t^{20}) is

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7.Binomial Theorem

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Coefficient of $t^{20}$ in the expansion of $(1 + t^2)^{10}(1 + t^{10})(1 + t^{20})$ is

A

$^{10}C_5 + 2$

B

$^{10}C_5$

C

$^{10}C_5 + 1$

D

None of these

Solution

$\left(1+\mathrm{t}^{2}\right)^{10}\left(1+\mathrm{t}^{10}+\mathrm{t}^{20}+\mathrm{t}^{30}\right)$

$ = \left( {1 + {\,^{10}}{{\rm{C}}_1}{{\rm{t}}^2} + {\,^{10}}{{\rm{C}}_2}{{\rm{t}}^4} + \ldots . + {\,^{10}}{{\rm{C}}_{10}}{{\rm{t}}^{20}}} \right)$

$\left( {1 + {t^{10}} + {t^{20}} + {t^{30}}} \right)$

$\therefore $ Coefficient $ = {\,^{10}}{{\rm{C}}_{10}} + {\,^{10}}{{\rm{C}}_5} + {\,^{10}}{{\rm{C}}_0} = 2 + {\,^{10}}{{\rm{C}}_5}$

Standard 11

Mathematics

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