The largest $\mathrm{n} \in \mathbf{N}$ such that $3^{\mathrm{n}}$ divides 50 ! is :

The term independent of $x$ in the expansion of $\left(\frac{(x+1)}{\left(x^{2 / 3}+1-x^{1 / 3}\right)}-\frac{(x-1)}{\left(x-x^{1 / 2}\right)}\right)^{10}, x>1$, is :

The remainder, when $7^{103}$ is divided by 23, is equal to:

The least value of n for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7}+\sqrt[12]{11})^n$ is 183, is :