The coefficient of $x^3$ in the expansion of $(x^2 - x + 1)^{10} (x^2 + 1)^{15}$ is equal to:

In the expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^n, n \in N$,if the ratio of the $15^{\text{th}}$ term from the beginning to the $15^{\text{th}}$ term from the end is $\frac{1}{6}$,then the value of ${}^n C_3$ is:

The coefficient of $x^{70}$ in $x^2(1+x)^{98} + x^3(1+x)^{97} + x^4(1+x)^{96} + \ldots + x^{54}(1+x)^{46}$ is ${}^{99}C_p - {}^{46}C_q$. Then a possible value of $p+q$ is:

The term independent of $x$ in the expansion of $\left(x-\frac{2}{\sqrt{x}}\right)^{21}$ is

$A$ person takes an examination consisting of four papers,each with a maximum of $m$ marks. The number of ways in which one can obtain a total of $2m$ marks is

If the expansion of $(y^2 + \frac{c}{y})^5$,the coefficient of $y$ is: