# What is the time complexity of this piece of code?

def find_perms(s):
"""
Finds permutations of the given string.
>>> find_perms("abc")
['cab', 'acb', 'abc', 'cba', 'bca', 'bac']
"""
if len(s) == 1:
return [s]
elif len(s) == 2:
return [s[0]+s[1], s[1]+s[0]]
else:
perm = find_perms(s[0:len(s)-1])
last_char = s[len(s)-1]
lst = []
for strings in perm:
for index in range(len(strings)+1):
string = strings[:index] + last_char + strings[index:]
lst.append(string)
return lst

I am trying to figure out the time complexity of this piece of code as I'm working on studying coding questions and their time complexity and I managed to solve this permutation problem using recursion as was required but I am wondering how do I accurately determine the big-$$O$$ time complexity for this as it is a mix of recursion followed by an iterative loop. I would really appreciate some helpful input on this.

Let $$f(n)$$ be the answer for an input of size $$n$$. then we have the following equation:
$$f(n) = f(n - 1) + (n - 1)! * n = f(n - 1) + n!$$
That's because you call the function for an input of size $$n - 1$$ which gives $$f(n - 1)$$, then you iterate over all permutations of size $$n - 1$$ which there are $$(n - 1)!$$ such permutations, and then you iterate over the position of the new element, which there are $$n$$ such positions. This gives us:
$$f(n) = [\sum_{i=1}^n i!] < (n + 1)!$$
You can easily prove this by induction. So $$f(n) = O((n + 1)!)$$.