Question

Evaluate fraction numerator P presubscript 4 subscript 2 times P presubscript 6 subscript 1 over denominator P presubscript 18 subscript 3 end fraction.A.1 over 68 B.begin display style 4 over 5 end style C.begin display style 3 over 68 end style D.begin display style 2 over 3 end style

Answer 1

Given:

`(4P_2\cdot6P_1)/(18P_3)`

Required:

We need to evaluate the given expression.

Step-by-step explanation:

Consider the formula.

`nP_r=(n!)/((n-r)!)`

Use this formula to evaluate the given expression.

`(4P_2\cdot6P_1)/(18P_3)=((4!)/((4-2)!)\cdot(6!)/((6-1)!))/((18!)/((18-3)!))`

`Use\text{ }((a)/(b))/((c)/(d))=(a)/(b)\cdot(d)/(c).`

`(4P_2\cdot6P_1)/(18P_3)=(4!)/((4-2)!)\cdot(6!)/((6-1)!)\cdot((18-3)!)/(18!)`

`(4P_2\cdot6P_1)/(18P_3)=(4!)/(2!)\cdot(6!)/(5!)\cdot(15!)/(18!)`

`(4P_2\cdot6P_1)/(18P_3)=(4*3*2!)/(2!)\cdot(6*5!)/(5!)\cdot(15!)/(18*17*16*15!)`

Cancel out the common factorials.

`(4P_2\cdot6P_1)/(18P_3)=(4*3)/(1)\cdot(6)/(1)\cdot(1)/(18*17*16)`

`(4P_2\cdot6P_1)/(18P_3)=(4*3*6)/(18*17*16)`

`(4P_2\cdot6P_1)/(18P_3)=(1*18)/(18*17*4)`

`(4P_2\cdot6P_1)/(18P_3)=(1)/(17*4)`

`(4P_2\cdot6P_1)/(18P_3)=(1)/(68)`

Final answer:

`(4P_2\cdot6P_1)/(18P_3)=(1)/(68)`