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

${16^{th}}$ term in the expansion of ${(\sqrt x - \sqrt y )^{17}}$ is

If the coefficients of ${5^{th}}$, ${6^{th}}$and ${7^{th}}$ terms in the expansion of ${(1 + x)^n}$be in $A.P.$, then $n =$

Show that the middle term in the expansion of $(1+x)^{2 n}$ is
$\frac{1.3 .5 \ldots(2 n-1)}{n !} 2 n\, x^{n},$ where $n$ is a positive integer.

If ${x^4}$ occurs in the ${r^{th}}$ term in the expansion of ${\left( {{x^4} + \frac{1}{{{x^3}}}} \right)^{15}}$, then $r = $

The coefficient of ${x^{ - 7}}$ in the expansion of ${\left( {ax - \frac{1}{{b{x^2}}}} \right)^{11}}$ will be