To solve this, we need to first know the general formulas for permutations and combinations:

#P_(n,k)=(n!)/((n-k)!); n="population", k="picks"#

#C_(n,k)=(n!)/((k)!(n-k)!)# with #n="population", k="picks"#

So let's set them equal to each other and insert what we know:

#P_(n,4)=30(C_(n-1,3))#

#(n!)/((n-4)!)=(30(n-1)!)/((3)!((n-1)-3)!)#

We can combine terms in the right hand side denominator:

#(n!)/((n-4)!)=(30(n-1)!)/((3)!(n-4)!)#

We can now multiply the left side top and bottom by #3!# to have our denominators match:

#(n!)/((n-4)!)((3!)/(3!))=(30(n-1)!)/((3)!(n-4)!)#

#(3!n!)/(3!(n-4)!)=(30(n-1)!)/((3)!(n-4)!)#

With the denominators the same, we can multiply through by it (and thus eliminate them) and thereby equate the numerators:

#3!n! =30(n-1)!#

We can rewrite the left side, recognizing that #n! =nxx(n-1)!#

#3!(nxx(n-1)!)=30(n-1)!#

#3!n(n-1)! =30(n-1)!#

divide through by #(n-1)!#:

#3!n =30#

#n=30/(3!)#

#color(blue)(ul(bar(abs(color(black)(n=30/6=5))))#

Let's check this:

#(5!)/((5-4)!)=(30(5-1)!)/((3)!((5-1)-3)!)#

#(5!)/(1!)=(30(4!))/((3)!((4)-3)!)#

#5! =(30(4!))/((3)!(1!)#

#5! =(30(4!))/6#

#5! =5(4!)#

#5! =5! color(white)(000)color(green)sqrt#