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  • The coefficient of  <img alt="\mathrm{x}^{53}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx

The coefficient of  \mathrm{x}^{53}  in \sum _{r}^{100} { }^{100} C_r(x-3)^{100-r} 2^r is

 

Option: 1

{ }^{100} \mathrm{C}_{51}


Option: 2

{ }^{100} \mathrm{C}_{52}


Option: 3

-{ }^{100} \mathrm{C}_{53}


Option: 4

{ }^{100} \mathrm{C}_{54}


Answers (1)

best_answer

\sum _{r}^{100} { }^{100} C_r(x-3)^{100-r} 2^r

=((x-3)+2)^{100}=(x-1)^{100}=(1-x)^{100} \\

\sum _{r}^{100}{ }^{100} C_r(-x)^r=\sum _{r}^{100}(-1)^r(-1)^r{ }^{100} C_r x^r

Coefficient of  x^{53}=(-1)^{53}\,\,{ }^{100} C_{53}=-{ }^{100} C_{53}.

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Riya

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