List of top Mathematics Questions on Binomial Expansion

In the expansion of \[ \left(9x-\frac{1}{3\sqrt{x}}\right)^{18}, \] if the coefficient of the term independent of \(x\) is \(221k\), then the value of \(k\) is:
Let \( \alpha_1 \) and \( \beta_1 \) be the distinct roots of \( 2x^2 + (\cos\theta)x - 1 = 0, \ \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha_1 + \beta_1 \), then \( 16(M + m) \) equals:

If \[ \sum_{r=0}^{10} \left( \frac{10^{r+1} - 1}{10^r} \right) \cdot {^{11}C_{r+1}} = \frac{\alpha^{11} - 11^{11}}{10^{10}}, \] then \( \alpha \) is equal to:

The number of integral terms in the expansion of $ \left( 5^{\frac{1}{2}} + 7^{\frac{1}{8}} \right)^{1016} $ is:
The number of rational terms in the binomial expansion of $ \left( 5^{1/2} + 7^{1/8} \right)^{1016} $ is:
If \( (1 + x + x^2)^{10} = 1 + a_1 x + a_2 x^2 + \dots \), then \( (a_1 + a_3 + a_5 + \dots + a_{19}) - 11a_2 \) equals to:
Let \( \alpha, \beta, \gamma \) and \( \delta \) be the coefficients of \( x^7, x^5, x^3, x \) respectively in the expansion of \( (x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, \, x > 1 \). If \( \alpha u + \beta v = 18 \), \( \gamma u + \delta v = 20 \), then \( u + v \) equals:

The minimum value of $ n $ for which the number of integer terms in the binomial expansion $\left(7^{\frac{1}{3}} + 11^{\frac{1}{12}}\right)^n$ is 183, is

If in the expansion of \( (1 + x)^p (1 - x)^q \), the coefficients of \( x \) and \( x^2 \) are 1 and -2, respectively, then \( p^2 + q^2 \) is equal to:

The sum of all rational terms in the expansion of \( \left( 1 + 2^{1/3} + 3^{1/2} \right)^6 \) is equal to