## Related questions with answers

In how many ways can we select a chairperson, vice-chairperson, secretary, and treasurer from a group of 12 persons?

Solution

VerifiedWe are interested in the number of 4 people from a set with 12 persons:

$\begin{align*} n&=12 \\ r&=4 \end{align*}$

The order in which we select the 4 people matters (as a different order of the people results in different chairperson, vice-chairperson, secretary and treasurer), thus we need to use the definition of a $\textbf{permutation}$.

A $\textbf{r-permutation}$ of a set of elements is an ordering of the $r$ of the elements in the set in a row. The number of $r$-permutations of a set of $n$ distinct objects is $P(n,r)=\frac{n!}{(n-r)!}$.

$\begin{align*} P(12,4)&=\frac{12!}{(11-4)!} \\ &=\frac{12!}{8!} \\ &=\frac{12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1} \\ &=12\cdot 11\cdot 10\cdot 9 \\ &=11,880 \end{align*}$

Thus there are 11,880 ways to select a chairperson, vice-chairperson, secretary, and treasurer from a group of 12 persons.

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