Expanding \((1+0.2)^{1000}_{}\) by the binomial theorem and doing no further manipulation gives $${1000 \choose 0}(0.2)^0+{1000 \choose 1}(0.2)^1+{1000 \choose 2}(0.2)^2+\cdots+{1000 \choose 1000}(0.2)^{1000}$$ $$= A_0 + A_1 + A_2 + \cdots + A_{1000},$$ where \(A_k = {1000 \choose k}(0.2)^k\) for \(k = 0,1,2,\ldots,1000\). For which \(k_{}^{}\) is \(A_k^{}\) the largest?

(第九届AIME1991 第3题)