Expanding \((1+0.2)^{1000}\) by the Binomial theorem gives

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\({1000 \choose 0}(0.2)^0+{1000 \choose 1}(0.2)^1+{1000 \choose 2}(0.2)^2+\cdots+{1000 \choose 1000}(0.2)^{1000}\) \(= A_0 + A_1 + A_2 + \cdots + A_{1000}\)

Where \(A_k = {1000 \choose k}(0.2)^k\) for \(k = 0,1,2,\ldots,1000\).

For which \(k_{}^{}\) is \(A_k^{}\) the largest?

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