election - how many possible outcomes _The question:_ In an association there are twelve members. Three persons shall be elected as chairman, deputy and clerk. How many outcomes are possible? _My reasoning:_ three out of twelve people will be elected ($12 * 11 * 10$). For these three people there are six possbilities to assign the roles ($3 * 2 * 1$). So in total there should be $12 * 11 * 10 * 3 * 2 * 1$ possible outcomes. I am unsure if the factor 6 is needed which accounts for the assignment of the roles to the elected people. One could also argue, that for the role of the chairman there are 12 people to choose from, for the deputy there are 11 and for the clerk there are 10, which would give only $12 * 11 * 10$. But then, this would restrict the order in which the roles are elected. Is this correct?

Your argument that the factor $6$ is not needed is correct. It does not restrict the order the roles are elected. For whichever office is considered first there are $12$ choices, then $11$ for the next office, and finally $10$ for the third. The factor $6$ would only be needed if you wanted to consider the election of A as chairman, B as deputy, C as clerk different from electing first C as clerk, then A as chairman, then B as deputy, which seems quite perverse.

If you want to choose three unordered people out of twelve, there are ${12 \choose 3}=\frac {12\cdot 11 \cdot 10}{3 \cdot 2 \cdot 1}$ ways. Now multiplying by $6$ lets you assign the order or the offices and gets you back to $12\cdot 11 \cdot 10$