If the coefficients of ${x^7}$ and ${x^8}$ in ${\left( {2 + \frac{x}{3}} \right)^n}$ are equal, then $n$ 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

The sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^{2}}\right)^{m}, x \neq 0, m$ being a natural number, is $559 .$ Find the term of the expansion containing $x^{3}$

The coefficients of three consecutive terms in the expansion of $(1+a)^{n}$ are in the ratio $1: 7: 42 .$ Find $n$

Let the coefficients of three consecutive terms in the binomial expansion of $(1+2 x)^{ n }$ be in the ratio $2: 5: 8$. Then the coefficient of the term, which is in the middle of these three terms, is $...........$.

If some three consecutive in the binomial expansion of ${\left( {x + 1} \right)^n}$ in powers of $x$ are in the ratio $2 : 15 : 70$, then the average of these three coefficient is