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#
