Greatest Binomial Coefficients

Q. A series, whose $n^{th}$ term is $\left(\frac{n}{x}\right)+y$, then sum of $r$ terms will be :

1. $\left\{\frac{r(r+1)}{2x}\right\}+ry$
2. $\left\{\frac{r(r-1)}{2x}\right\}$
3. $\left\{\frac{r(r+1)}{2x}\right\}-rx$
4. ${\left\{\frac{r(r-1)}{2x}\right\}}^2-xy$

Q. The Greatest co-efficient in the expansion of $(1+x{)}^{2n+2}$ is

1. $\frac{\left(2n\right)!}{(n!{)}^2}$
2. $\frac{(2n+2)!}{\left(\right(n+1)!{)}^2}$
3. $\frac{(2n+2)!}{n!(n+1)!}$
4. $\frac{\left(2n\right)!}{n!(n+1)!}$

Q. If the coefficient of $5^{\text{th}}$ term is numerically the greatest coefficient in the expansion of $(1-x{)}^n,$ then the positive integral value of $n$ is

1. $9$
2. $8$
3. $7$
4. $10$

Q. Number of rational terms in the expansion of ${(\sqrt{2}+\sqrt[5]{54})}^{100}$ is

Q. The coefficient of $\frac{1}{y^2}$ in ${(y+\frac{c^3}{y^2})}^{10}$ is $3360$. If the value of $y=2$, then

1. $c=\sqrt[3]{32}$
2. greatest term will be $2880$
3. $5^{\text{th}}$ term will be coefficient of $y^{-2}$
4. $3^{\text{rd}}$ term will have greatest numerical value