Binomial Theore...

Find the middle terms in the expansion of \(\left(2 \mathrm{x}+\frac{1}{\mathrm{x}}\right)^{8}\).

Evaluate: \(\sum_{\mathrm{r}=0}^{\mathrm{r}=10}\left(\mathrm{r} \times{ }^{10} \mathrm{C}_{\mathrm{r}}\right)\)

The \(6^{\text {th }}\) term in the expansion of \(\left[\frac{1}{x^{\frac{8 }{ 3}}}+x^{2} \log _{10} \mathrm{x}\right]^{8}\) is 5600 , then \(\mathrm{x}\) is equal to:

The coefficient of of the middle term in the binomial expansion in \(x\) of \((1+\beta x)^{4}\) and of \((1-\beta x)^{6}\) is same, if \(\beta\) equals:

In an examination a candidate has to pass in each of the 5 subjects. In how many ways he can fail?

If \(\mathrm{T}_{\mathrm{r}}\) denotes the \(\mathrm{r}^{\text {th }}\) term in the expansion of \(\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)^{23}\), then:

Sum of the power of all coefficient's in the expansion of \(\left(2 x^{2}-\frac{5}{x}\right)^{13}\) is:

The \(9^{\text {th }}\) term from the end in \((x-\frac{1 }{ x})^{12}\) is:

The coefficient of the term independent of \(x\) in the expansion of \(\left(\sqrt{\frac{x}{3}}+\frac{3}{2 x^{2}}\right)^{10}\) is:

Find the \(9^{\text {th }}\) term in the expansion of \(\left(\frac{a}{b}-\frac{b}{2 a^{2}}\right)^{12}\).