#GAP DERANGEMENT DEMONSTRATION:
# WE WILL USE GAP TO COUNT THE NUMBER OF DERANGEMENS First lets list the derangements of 4 elements:

n:=4;
	G:=SymmetricGroup(n);    #The Symmetric Group is all permutations of 1,2,...n. It has n! elements.
	t:=0;
		for g in G do
			if NrMovedPoints(g)=n then #NrMovedPoints counts how many points are moved, derangements are those that move n
		Print(ListPerm(g), "\n");  t:=t+1;
			else fi;
		od;

	Print("t=", t); Print(","); Print(" The fraction of deragnments for permutations of ", n, " elements is  ", Float(t/Factorial(n)));




Now 6 elements:


n:=8;
	G:=SymmetricGroup(n);
	t:=0;
		for g in G do
			if NrMovedPoints(g)=n then
				Print(ListPerm(g), "\n");  t:=t+1;
			else fi;
		od;
Print("t=", t);; Print(",");; Print(" The fraction of deragnments for permutations of ", n, " elements is  ", Float(t/Factorial(n)));;




# Now lets loop through lots of values of n, note that ;; makes GAP suppress ouput. We won't ask it to print the derangements, just to count them, and we just print out the fraction.


for n in [3..12] do

	G:=SymmetricGroup(n);;

		t:=0;;
	for g in G do
		if NrMovedPoints(g)=n then
			t:=t+1;;
		else fi;
	od;;

Print(" The fraction of derangnments for permutations of ", n, " elements is  ", Float(t/Factorial(n)), "\n");

		od;