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"""

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 factorial(n) // (factorial(k) * factorial(n - k))

if __name__ == "__main__":

print(

"The number of five-card hands possible from a standard",

f"fifty-two card deck is: {combinations(52, 5)}\n",

)

print(

"If 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.",

)