Given three remainders, what is the function?

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A certain function of \(x\) can be expressed in the form \[a+b(x-1)+c(x-1)(x-2)+2(x-1)(x-2)(x-3),\] where \(a\), \(b\), \(c\), are numerical. When the function is divided in turn by \(x-1\), \(x-2\), \(x-3\) the remainders are \(10\), \(3\), \(4\). Calculate the values of \(a\), \(b\), \(c\).