Question

I was trying to get help earlier but my internet went out

Answer 1

We have two expressions of permutations and combinations. We have to expand and simplify the expression:

$\frac{_8P_5}{_8C_5}=\frac{P(8,5)}{C(8,5)}$

We can expand the permutation as:

$\begin{gathered} P(n,r)=\frac{n!}{(n-r)!} \\ P(8,5)=\frac{8!}{(8-5)!}=\frac{8!}{3!} \end{gathered}$

and the combination as:

$\begin{gathered} C(n,r)=\frac{n!}{r!(n-r)!} \\ C(8,5)=\frac{8!}{5!(8-3)!}=\frac{8!}{5!3!} \end{gathered}$

Then, combining the two as:

$\frac{P(8,5)}{C(8,5)}=\frac{\frac{8!}{3!}}{\frac{8!}{5!3!}}=\frac{1}{5!}=\frac{1}{120}$

Answer: the quotient is equal to 1/120.