Calvin thinks of a 5-digit number. The product of the digits is 200.

How many different numbers which Calvin could have thought of?

The answer is 100.

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We note that $200 = 2^3\times 5^2$ . There are only $3$ possible combinations and the respective permutations are as follows:

$\begin{array} {rrrr} \text{Combination} & & \text{Permutations} \\ 11558 & \Rightarrow & \dfrac{5!}{2!\times 2!} = \dfrac{120}{4} & = 30 \\ 12455 & \Rightarrow & \dfrac{5!}{2!} = \dfrac{120}{2} & = 60 \\ 22255 & \Rightarrow & \dfrac{5!}{2!\times 3!} = \dfrac{120}{12} & = 10 \\ \hline & & \text{Total:} & 100 \end{array}$

The total number of such 5-digit numbers is $\boxed{100}$ .