Parlour Magician

In a parlour game, the “magician” asks one of the participants to
think of a three-digit number \({ abc }_{ 10 }\). Then the magician asks the participant to add the five numbers \({ acb }_{ 10 }\), \({ bca }_{ 10 }\), \({ bac }_{ 10 }\), \({ cab }_{ 10 }\) and \({ cba }_{ 10 }\), and reveal their sum. Suppose the sum was \(3194\). What was \({ abc }_{ 10 }\) originally?

Details and Assumptions

\({ abc }_{ 10 }\) means a 3-digit number with \(a,b,c\) as it's digits and not \(a\times b\times c\).

This problem is not original.

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