Question

There are 21 students in a homeroom. How may different ways can they be chosen tobe elected President, Vice President, Treasurer, and Secretary?

Answer 1

Permutations

A permutation is a mathematical technique that determines the number of possible arrangements in a set when the order of the arrangements matters.

We have a total of 21 students to choose from. There are four positions to be filled.

The total number of ways to fill the positions out of the 21 students can be calculated by using the permutation formula:

$_nP_r=\frac{n!}{(n-r)!}$

Where n is the total number of elements in the set and r is the number of elements selected. In our case, n=21 and r = 4, thus:

$_{21}P_4=\frac{21!}{(21-4)!}$

Applying the properties of the factorial function:

$\begin{gathered} _{21}P_4=\frac{21!}{(17)!} \\ _{21}P_4=\frac{21\cdot20\cdot19\cdot18\cdot17!}{(17)!} \end{gathered}$

Simplifying:

$\begin{gathered} _{21}P_4=21\cdot20\cdot19\cdot18 \\ _{21}P_4=143,640 \end{gathered}$

There are 143,640 possible ways to choose for the 4 positions