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A memory chip is being designed to hold a number of transistors and heat sinks. The transistors hold memory while the heat sinks cool the chip. There must be at least one heat sink for every $2$,$000$ transistors to prevent overheating. Also, each transistor has an area of $2.0 \times 10^{-10} \mathrm{~mm}^{2}$ (square millimeters), each heat sink has an area of $3.6 \times 10^{-6} \mathrm{~mm}^{2}$, and the total area of transistors and heat sinks must be at most $2 \mathrm{~mm}^{2}$. What is the approximate maximum number of transistors that the chip can hold according to this design?$\newline$Choose $1$ answer:$\newline$(A) $2.78 \times 10^{2}$$\newline$(B) $5.56 \times 10^{5}$$\newline$(C) $1.0 \times 10^{9}$$\newline$(D) $1.0 \times 10^{10}$