If the co-efficient of the 5th term is divided by the co-efficient of the 13th terms in the expansion of \left ( \frac{3}{2}x-\frac{2}{3}y \right )^{16}, then the quotient is

  • Option 1)

    (\frac{2}{3})^{16}

  • Option 2)

    (\frac{2}{3})^{8}

  • Option 3)

    (\frac{3}{2})^{8}

  • Option 4)

    (\frac{3}{2})^{16}

 

Answers (1)

As we learnt in 

General Term in the expansion of (x+a)^n -

T_{r+1}= ^{n}c_{r}\cdot x^{n-r}\cdot a^{r}
 

- wherein

Where r\geqslant 0 \, and \, r\leqslant n

r= 0,1,2,----n

 

 \left ( \frac{3}{2}x- \frac{2}{3}y\right )^{16}\\*T_{5}=^{16}C_{4}\left ( \frac{3}{2}\right )^{12}\left ( \frac{-2}{3}\right )^{4}\\*T_{13}=^{16}C_{12} \left ( \frac{3}{2}\right )^{4} \left ( \frac{-2}{3}\right )^{12}

Coeffecient of T_{5} is divided by T_{13}

\frac{^{16}C_{4}\left ( \frac{3}{2}\right )^{12}\left ( \frac{-2}{3}\right )^{4}}{^{16}C_{12} \left ( \frac{3}{2}\right )^{4} \left ( \frac{-2}{3}\right )^{12}}=\left ( \frac{3}{2} \right )^{16}


Option 1)

(\frac{2}{3})^{16}

Incorrect

Option 2)

(\frac{2}{3})^{8}

Incorrect

Option 3)

(\frac{3}{2})^{8}

Incorrect

Option 4)

(\frac{3}{2})^{16}

Correct

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