Solution:
The permutation formula is expressed as
[tex]\begin{gathered} P^n_r=\frac{n!}{(n-r)!} \\ \end{gathered}[/tex]
The combination formula is expressed as
[tex]\begin{gathered} C^n_r=\frac{n!}{(n-r)!r!} \\ \\ \end{gathered}[/tex]
where
[tex]\begin{gathered} n\Rightarrow total\text{ number of objects} \\ r\Rightarrow number\text{ of object selected} \end{gathered}[/tex]
Given that 6 objects are taken at a time from 8, this implies that
[tex]\begin{gathered} n=8 \\ r=6 \end{gathered}[/tex]
Thus,
Number of permuations:
[tex]\begin{gathered} P^8_6=\frac{8!}{(8-6)!} \\ =\frac{8!}{2!}=\frac{8\times7\times6\times5\times4\times3\times2!}{2!} \\ 2!\text{ cancel out, thus we have} \\ \begin{equation*} 8\times7\times6\times5\times4\times3 \end{equation*} \\ \Rightarrow P_6^8=20160 \end{gathered}[/tex]
Number of combinations:
[tex]\begin{gathered} C^8_6=\frac{8!}{(8-6)!6!} \\ =\frac{8!}{2!\times6!}=\frac{8\times7\times6!}{6!\times2\times1} \\ 6!\text{ cancel out, thus we have} \\ \frac{8\times7}{2} \\ \Rightarrow C_6^8=28 \end{gathered}[/tex]
Hence, there are 28 combinations and 20160 permutations.
