What makes a number binomial? Finale

Discrete Mathematics Level 3

From the binomial expansion of $$(1 + 0.2)^{1000} = \sum_{k = 0}^{1000}A_{k},$$

when $$A_{k} = \binom{1000}{k}(0.2)^{k},$$

find the value of $$k$$ where the maximum value of $$A_{k}$$ is reached.

Credit goes to an AIME problem.

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