The greatest coefficient in the expansion of $\left ( 1+x \right )^{2n+2}$ is

The correct option is $\mathbf{B} \frac{(2 n+2)!}{[(n+1)!]^{2}}$

The greatest term of the binomial expansion of $(1+\mathrm{x})^{2 \mathrm{n}+2}$ will be the middle term.

Since in the above case there will be in total $2 \mathrm{n}+3$ terms, hence there will be a single middle term which is

$\begin{array}{l} \mathrm{T}_{\frac{2 \mathrm{n}+2}{2}}^{2} \\ =\mathrm{T}_{\mathrm{n}+2} \\ =\mathrm{T}_{(\mathrm{n}+1)+1} \end{array}$

Now we know that the expression for general term is

$\mathrm{T}_{\mathrm{r}+1}={ }^{2 \mathrm{n}+2} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{\mathrm{r}} \text {. }$

Substituting $\mathrm{r}=\mathrm{n}+1$ gives the coefficient as

$={ }^{2 n+2} C_{n+1}$