Intuitive idea: in , all factors are equal to 5, whereas in , the factors start with 1 but soon overtake 5 and continue to increase. So, , but this won't last long. Calculate and for n=6,7,...,12. As you said, is already less than 12!. This can be established by direct calculation (e.g., in Windows' calculator).1) 5^n is O(n!)

Now what happens when we move from to for ? is multiplied by 5 to get ; but is multiplied by to get . So, it is clear that implies for .