## Related questions with answers

a. In how many ways can a club with 20 members choose a president and a vice president?

b. In how many ways can the club choose a 2-person governing council?

Solutions

Verifieda) Since the order (or positions) matter, then permutation is used. With two positions to fill up, then the 20 members can be arranged in:

$\begin{align*} _{20}P_2 &= \dfrac{20!}{(20-2)!} \\ &= \dfrac{20 \cdot 19 \cdot \cancel{18!}}{ \cancel{18!} } \\ &= 380 \text{ ways}. \\ \end{align*}$

Note the difference between permutations and combinations:

$\color{#4257b2}$

${\bf Permutation}$: a selection of r members from a set of n members, in which $\fbox{{\bf order is important}}$.

(Arrangement, itinerary, list,...)

${}_{n}P_{r}=P(n,r)=\displaystyle \frac{n!}{(n-r)!}=n(n-1)\cdot...\cdot(n-r+1)\\\\$

{\bf Combination}: a selection of r members from a set of n members, in which \fbox{{\bf order is not important}}.\
(Subset, group, committee, )\${}*{n}C*{r}=C(n,r)=\displaystyle $\frac{n!}{(n-r)!\cdot r!}$=$$\frac{n(n-1)\cdot...\cdot(n-r+1)}{1\cdot 2\cdot...\cdot r}$

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