Compute $(98)^{5}$.

If $a_k$ is the coefficient of $x^k$ in the expansion of $(1+x+x^2)^n$ for $k=0, 1, 2, \ldots, 2n$,then $a_1+2a_2+3a_3+\ldots+2na_{2n}$ is equal to

The sum of all the coefficients in the binomial expansion of $(x^2 + x - 3)^{319}$ is

Let $n$ be a positive integer. Let $A = \sum_{k=0}^{n} (-1)^{k} {}^{n}C_{k} \left[ \left(\frac{1}{2}\right)^{k} + \left(\frac{3}{4}\right)^{k} + \left(\frac{7}{8}\right)^{k} + \left(\frac{15}{16}\right)^{k} + \left(\frac{31}{32}\right)^{k} \right]$. If $63A = 1 - \frac{1}{2^{30}}$,then $n$ is equal to ......

The coefficient of $x^4$ in the expansion of $(1 + x + x^2 + x^3)^n$ is

In the expansion of $(1 + x + x^3 + x^4)^{10}$,the coefficient of $x^4$ is