Answer :
The number of students in a homeroom is
[tex]n=19[/tex]The number of posts to be chosen from is
[tex]r=3[/tex]Concept:
The above selection can be done using the permutation formula below
[tex]^nP_r=\frac{n!}{(n-r)!}[/tex]By substituting the values, we will have
[tex]\begin{gathered} ^nP_r=\frac{n!}{(n-r)!} \\ ^{19}P_3=\frac{19!}{(19-3)!} \\ ^{19}P_3=\frac{19!}{16!} \end{gathered}[/tex]By expanding the factorial, we will have
[tex]\begin{gathered} ^{19}P_3=\frac{19!}{16!} \\ ^{19}P_3=\frac{19\times18\times17\times16!}{16!} \\ ^{19}P_3=19\times18\times17 \\ ^{19}P_3=5814\text{ ways} \end{gathered}[/tex]Hence,
The final answer is = 5,814 ways