```
"""
https://en.wikipedia.org/wiki/Combination
"""
from math import factorial
def combinations(n: int, k: int) -> int:
"""
Returns the number of different combinations of k length which can
be made from n values, where n >= k.
Examples:
>>> combinations(10,5)
252
>>> combinations(6,3)
20
>>> combinations(20,5)
15504
>>> combinations(52, 5)
2598960
>>> combinations(0, 0)
1
>>> combinations(-4, -5)
...
Traceback (most recent call last):
ValueError: Please enter positive integers for n and k where n >= k
"""
# If either of the conditions are true, the function is being asked
# to calculate a factorial of a negative number, which is not possible
if n < k or k < 0:
raise ValueError("Please enter positive integers for n and k where n >= k")
return int(factorial(n) / ((factorial(k)) * (factorial(n - k))))
if __name__ == "__main__":
print(
"\nThe number of five-card hands possible from a standard",
f"fifty-two card deck is: {combinations(52, 5)}",
)
print(
"\nIf a class of 40 students must be arranged into groups of",
f"4 for group projects, there are {combinations(40, 4)} ways",
"to arrange them.\n",
)
print(
"If 10 teams are competing in a Formula One race, there",
f"are {combinations(10, 3)} ways that first, second and",
"third place can be awarded.\n",
)
```