The combination formula is
#C_(n,k)=(n!)/((k!)(n-k)!)# with #n="population", k="picks"#
We're being asked to find a value #x# such that:
#C_(x,4)=5C_(x-2,3)#
#(x!)/((4!)(x-4)!)=((5)(x-2)!)/((3!)(x-2-3)!)#
#(x!)/((24)(x-4)!)=((5)(x-2)!)/((6)(x-5)!)#
#(x (x-1) (x-2)(x-3)(x-4)!)/((24)(x-4)!)=((5)(x-2)(x-3)(x-4)(x-5)!)/((6)(x-5)!)#
#(x (x-1) (x-2)(x-3))/(24)=((5)(x-2)(x-3)(x-4))/(6)#
#(6)x (x-1) (x-2)(x-3)=(24)(5)(x-2)(x-3)(x-4)#
#x (x-1) =(4)(5)(x-4)#
#x^2-x =20x-80#
#x^2-21x+80 =0#
#(x-5)(x-16)=0#
#x=5, 16#
Checking the answer:
#(5!)/((4!)(5-4)!)=((5)(5-2)!)/((3!)(5-2-3)!)#
#5=((5)(3!))/((3!)(0!))=5color(white)(000)color(green)root#
#(16!)/((4!)(16-4)!)=((5)(16-2)!)/((3!)(16-2-3)!)#
#(16!)/((4!)(12!))=((5)(14!))/((3!)(11!))#
#(16xx15xx14xx13)/24=((5)(14xx13xx12))/6#
#(16xx15xx14xx13)=(20)(14xx13xx12)#
#43,680=43,680color(white)(000)color(green)root#
