Binomial Theore...

Consider the expansion of \((8 x+2 y)^{23}\). Find the ratio between the eighth and the seventh terms.

If \(a_{1}, a_{2}, a_{3}\) and \(a_{4}\) are 4 consecutive terms in the expansion of \((1+x)^{n}\), then \(\frac{a_{1}}{a_{1}+a_{2}}+\frac{a_{3}}{a_{3}+a_{4}}=\) ?

In the expansion of \(\left(x^{3}-\frac{1}{x^{2}}\right)^{15}\), the constant term, is:

Find the coefficients of the term independent of \(x\) in the expansion of \(\left(\frac{x^{3 / 2}-1}{x+1+x^{1 / 2}}-\frac{x^{3 / 2}+1}{x+1-x^{1 / 2}}\right)^{5}\).

In the expansion of \((1+a x)^{n}\), the first three terms are respectively \(1,12 \mathrm{x}\) and \(64 \mathrm{x}^{2}\). What is \(\mathrm{n}\) equal to:

If \(\left(1+x-2 x^{2}\right)^{6}=1+a_{1} x+a_{2} x^{2}+\ldots \ldots+a_{12} x^{12}\), then the expression \(a_{2}+\) \(a_{4}+a_{6}+\ldots \ldots+a_{12}\) has the value:

The number of terms in the expansion of \(\left(1+3 x+3 x^{2}+x^{3}\right)^{6}\) is:

In the binomial expansion of \((a+b)^{n}\), the coefficients of the \(3^{\text {rd }}\) and \(15^{\text {th }}\) terms are equal to each other. Find value of \(n\) :

Find the coefficient of \(x^{4}\) in the expansion of \(\left(1+x+x^{2}+x^{3}\right)^{11}\).

The smallest natural number \(n\), such that the coefficient of \(x\) in the expansion of \(\left(x^{2}+\frac{1}{x^{3}}\right)^{n}\) is \({ }^{n} C_{23}\), is: