200+ followers (easy version)

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

How many different numbers which Calvin could have thought of?


Try the harder version at here


The answer is 100.

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1 solution

Chew-Seong Cheong
Sep 17, 2015

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

Combination Permutations 11558 5 ! 2 ! × 2 ! = 120 4 = 30 12455 5 ! 2 ! = 120 2 = 60 22255 5 ! 2 ! × 3 ! = 120 12 = 10 Total: 100 \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 100 \boxed{100} .

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