The number of integral terms in the expansion of ${({5^{1/2}} + {7^{1/6}})^{642}}$ is

Coefficient of $t^{20}$ in the expansion of $(1 + t^2)^{10}(1 + t^{10})(1 + t^{20})$ is

In the expansion of ${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$, the coefficient of ${x^4}$is

The coefficient of ${t^{24}}$ in the expansion of ${(1 + {t^2})^{12}}(1 + {t^{12}})\,(1 + {t^{24}})$ is

In the expansion of ${\left( {2{x^2} - \frac{1}{x}} \right)^{12}}$, the term independent of x is

If coefficients of ${(2r + 1)^{th}}$ term and ${(r + 2)^{th}}$ term are equal in the expansion of ${(1 + x)^{43}},$ then the value of $r$ will be