We have the following expression:
[tex]\frac{6!5!}{8!3!}[/tex]
And we need to evaluate the expression and simplify the answer as much as possible.
1. To find the resulting expression, we can see that we have the factorial of some numbers, and we need to remember that:
[tex]\begin{gathered} n!\text{ is the product of the sequence of whole numbers that precede n by n. That is:} \\ \\ 3!=3*2*1=6 \\ \\ 4!=4*3*2*1=24,\text{ and so forth.} \end{gathered}[/tex]
2. Then we have that the expression can be simplified as follows:
[tex]\begin{gathered} \frac{6!5!}{8!3!}=\frac{6!*5!}{8*7*6!*3!}\Rightarrow\frac{6!}{6!}=1,\text{ then we have:} \\ \\ \frac{6!5!}{8!3!}=\frac{5!}{8*7*3!}=\frac{5*4*3!}{8*7*3!}\Rightarrow\frac{3!}{3!}=1 \\ \\ \end{gathered}[/tex]
3. Then we finally have that:
[tex]\begin{gathered} \frac{6!5!}{8!3!}=\frac{5*4}{8*7}=\frac{5*4}{2*4*7}\Rightarrow\frac{4}{4}=1 \\ \\ \frac{6!5!}{8!3!}=\frac{5}{2*7}=\frac{5}{14} \\ \\ \frac{6!5!}{8!3!}=\frac{5}{14} \end{gathered}[/tex]
Therefore, in summary, the result is:
[tex]\frac{6!5!}{8!3!}=\frac{5}{14}[/tex]
