HOTS Zone : Koefisien Binomial

\(\begin{aligned} S&=(x-2)^4+8(x-2)^3+24(x-2)^2+32(x-2)+16\\ &=(x-2)^4+4\cdot2(x-2)^3+6\cdot2^2(x-2)^2+4\cdot2^3(x-2)+2^4\\ &=(x-2+2)^4\\ &=\color{blue}\boxed{\boxed{\color{black}x^4}} \end{aligned}\)

Perhatikan bahwa $$(x+y)^n=\sum_{i=0}^{n}\binom{n}{i}x^iy^{n-i}$$ Jika n = 2011, x = −1, y = 1, \begin{aligned} (-1+1)^{2011}&=\sum_{i=0}^{2011}\binom{2011}{i}(-1)^i(1)^{2011-i}\\ 0&=\sum_{i=0}^{2011}(-1)^i\binom{2011}{i}\\ \sum_{i=0}^{2011}(-1)^i\binom{2011}{i}&=\color{blue}\boxed{\boxed{\color{black}0}} \end{aligned}