The value of $4 \{^nC_1 + 4 \cdot ^nC_2 + 4^2 \cdot ^nC_3 + \dots + 4^{n-1} \cdot ^nC_n\}$ is:

Expand using Binomial Theorem $\left(1+\frac{x}{2}-\frac{2}{x}\right)^{4}, x \neq 0$.

For $|x| < 1$,the constant term in the expansion of $\frac{1}{(x-1)^2(x-2)}$ is

If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected,then $\frac{(1 + x)^{3/2} - (1 + \frac{1}{2}x)^3}{(1 - x)^{1/2}}$ may be approximated as

Find $(x+1)^{6}+(x-1)^{6}$. Hence or otherwise,evaluate $(\sqrt{2}+1)^{6}+(\sqrt{2}-1)^{6}$.

For $2 \le r \le n$,the expression $\binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2}$ is equal to: