Given:
[tex]\frac{4P_2\cdot6P_1}{18P_3}[/tex]Required:
We need to evaluate the given expression.
Explanation:
Consider the formula.
[tex]nP_r=\frac{n!}{(n-r)!}[/tex]Use this formula to evaluate the given expression.
[tex]\frac{4P_2\cdot6P_1}{18P_3}=\frac{\frac{4!}{(4-2)!}\cdot\frac{6!}{(6-1)!}}{\frac{18!}{(18-3)!}}[/tex][tex]Use\text{ }\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}\cdot\frac{d}{c}.[/tex][tex]\frac{4P_2\cdot6P_1}{18P_3}=\frac{4!}{(4-2)!}\cdot\frac{6!}{(6-1)!}\cdot\frac{(18-3)!}{18!}[/tex][tex]\frac{4P_2\cdot6P_1}{18P_3}=\frac{4!}{2!}\cdot\frac{6!}{5!}\cdot\frac{15!}{18!}[/tex][tex]\frac{4P_2\cdot6P_1}{18P_3}=\frac{4\times3\times2!}{2!}\cdot\frac{6\times5!}{5!}\cdot\frac{15!}{18\times17\times16\times15!}[/tex]Cancel out the common factorials.
[tex]\frac{4P_2\cdot6P_1}{18P_3}=\frac{4\times3}{1}\cdot\frac{6}{1}\cdot\frac{1}{18\times17\times16}[/tex][tex]\frac{4P_2\cdot6P_1}{18P_3}=\frac{4\times3\times6}{18\times17\times16}[/tex][tex]\frac{4P_2\cdot6P_1}{18P_3}=\frac{1\times18}{18\times17\times4}[/tex][tex]\frac{4P_2\cdot6P_1}{18P_3}=\frac{1}{17\times4}[/tex][tex]\frac{4P_2\cdot6P_1}{18P_3}=\frac{1}{68}[/tex]Final answer:
[tex]\frac{4P_2\cdot6P_1}{18P_3}=\frac{1}{68}[/tex]