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def binomial_coefficient(n: int, r: int) -> int:

"""

Find binomial coefficient using Pascal's triangle.

Calculate C(n, r) using Pascal's triangle.

:param n: The total number of items.

:param r: The number of items to choose.

:return: The binomial coefficient C(n, r).

>>> binomial_coefficient(10, 5)

252

>>> binomial_coefficient(10, 0)

>>> binomial_coefficient(0, 10)

>>> binomial_coefficient(10, 10)

>>> binomial_coefficient(5, 2)

>>> binomial_coefficient(5, 6)

>>> binomial_coefficient(3, 5)

>>> binomial_coefficient(-2, 3)

Traceback (most recent call last):

...

ValueError: n and r must be non-negative integers

>>> binomial_coefficient(5, -1)

Traceback (most recent call last):

...

ValueError: n and r must be non-negative integers

>>> binomial_coefficient(10.1, 5)

Traceback (most recent call last):

...

TypeError: 'float' object cannot be interpreted as an integer

>>> binomial_coefficient(10, 5.1)

Traceback (most recent call last):

...

TypeError: 'float' object cannot be interpreted as an integer

"""

if n < 0 or r < 0:

raise ValueError("n and r must be non-negative integers")

if 0 in (n, r):

return 1

c = [0 for i in range(r + 1)]

# nc0 = 1

c[0] = 1

for i in range(1, n + 1):

# to compute current row from previous row.

j = min(i, r)

while j > 0:

c[j] += c[j - 1]

j -= 1

return c[r]

if __name__ == "__main__":

from doctest import testmod

testmod()

print(binomial_coefficient(n=10, r=5))

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