$(102)^4 = ?$

Using the Binomial Theorem,evaluate $(101)^{4}$.

The coefficient of $x^{4}$ in the expansion of $(1+x+x^{2})^{10}$ is

If the number of terms in the expansion of $(x - 2y + 3z)^n$ is $45$,then $n=$

If $a$ and $b$ are distinct integers,prove that $a-b$ is a factor of $a^{n}-b^{n}$,whenever $n$ is a positive integer.

Find the value of $\frac{18^3 + 7^3 + 3 \times 18 \times 7 \times 25}{3^6 + 6 \times 243 \times 2 + 15 \times 81 \times 4 + 20 \times 27 \times 8 + 15 \times 9 \times 16 + 6 \times 3 \times 32 + 64}$