Question

2) The debate team with 18 members is to choose four officers: captain, co-captain, treasurer, and secretary. How many ways can those officers be selected?

Answer 1

We have 18 members that can be positioned in 4 positions.

As the positions are different from each other, the order matters.

Then, this is a permutation of 18 elements in 4 places with no repetition.

The number of permutations can be calculated as:

$\begin{gathered} P(n,r)=\frac{n!}{(n-r)!} \\ P(18,4)=\frac{18!}{(18-4)!}=\frac{18!}{14!}=18\cdot17\cdot16\cdot15=73440 \end{gathered}$

We could have derived this by doing this analysis:

We have 18 to choose for captain.

Then, we have 17 left to choose for co-captain.

Finally 16 can be chosen for treasurer and 15 are left for secretary.

Then, the posible teams are 18*17*16*15=73440.

Answer: we can select them in 73,440 ways.