Assuming that $$(1-2x)^7=a_0 + a_1x+a_2x^2+\cdots+a_7x^7$$
Find the value of
- $S_1 = a_1+a_2+\cdots + a_7$
- $S_2 = a_1+a_3+a_5+a_7$
- $S_3 = a_0+a_2+a_4+a_6$
- $S_4 = \mid a_0\mid + \mid a_1\mid +\cdots + \mid a_7\mid$
Show that $$\sum_{k=0}^n\left(2^k\binom{n}{k}\right)=3^n$$
Given $$P(x)=(1+x+x^2)^{100}=a_0+a_1x+\cdots+a_{200}x^{200}$$
Compute the following sums:
- $S_1=a_0+a_1+a_2+a_3 +\cdots+a_{200}$
- $S_2=a_0+a_2+a_4+a_6 +\cdots+a_{200}$.
If the sum of all coefficients in the expanded form of $(3x+1)^n$ is $256$, find the coefficient of $x^2$.
Find the value of $$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\binom{n}{3}+\cdots +(-1)^n\binom{n}{n}$$