



'''simple docstring'''
def A__    (     UpperCAmelCase_ = 2_0_0							):
						_UpperCamelCase					:		str 				=   [1, 2, 5, 1_0, 2_0, 5_0, 1_0_0, 2_0_0]
						_UpperCamelCase					:		List[Any] 				=   [0] * (pence + 1)
						_UpperCamelCase					:		Any 				=   1  # base case: 1 way to make 0 pence

						for coin in coins:
												for i in range(UpperCAmelCase_  ,  pence + 1  ,  1							):
																		number_of_ways[i] += number_of_ways[i - coin]
						return number_of_ways[pence]


if __name__ == "__main__":
					assert solution(200) == 73682






"""simple docstring"""
from __future__ import annotations

from scipy.special import comb  # type: ignore



class 			_UpperCAmelCase :






		'''simple docstring'''

		def __init__(self				,       a_					):





			'''simple docstring'''

			__snake_case	:						List[str]						=				list_of_points
			# Degree determines the flexibility of the curve.
			# Degree = 1 will produce a straight line.
			__snake_case	:						Optional[Any]						=				len(a_					) - 1



		def 				SCREAMING_SNAKE_CASE  (self				,       a_					):





			'''simple docstring'''

			assert 0 <= t <= 1, "Time t must be between 0 and 1."
			__snake_case	:						list[float]						=				[]
			for i in range(len(self.list_of_points					)					):
				# basis function for each i
				output_values.append(
				    comb(self.degree				,       a_					) * ((1 - t) ** (self.degree - i)) * (t**i)					)
			# the basis must sum up to 1 for it to produce a valid Bezier curve.
			assert round(sum(a_					)				,       5					) == 1
			return output_values



		def 				SCREAMING_SNAKE_CASE  (self				,       a_					):





			'''simple docstring'''

			assert 0 <= t <= 1, "Time t must be between 0 and 1."

			__snake_case	:						List[str]						=				self.basis_function(a_					)
			__snake_case	:						str						=				0.0
			__snake_case	:						Union[str, Any]						=				0.0
			for i in range(len(self.list_of_points					)					):
				# For all points, sum up the product of i-th basis function and i-th point.
				x += basis_function[i] * self.list_of_points[i][0]
				y += basis_function[i] * self.list_of_points[i][1]
			return (x, y)



		def 				SCREAMING_SNAKE_CASE  (self				,       a_ = 0.01					):





			'''simple docstring'''

			from matplotlib import pyplot as plt  # type: ignore

			__snake_case	:						list[float]						=				[]  # x coordinates of points to plot
			__snake_case	:						list[float]						=				[]  # y coordinates of points to plot

			__snake_case	:						int						=				0.0
			while t <= 1:
				__snake_case	:						Union[str, Any]						=				self.bezier_curve_function(a_					)
				to_plot_x.append(value[0]					)
				to_plot_y.append(value[1]					)
				t += step_size

			__snake_case	:						List[Any]						=				[i[0] for i in self.list_of_points]
			__snake_case	:						Any						=				[i[1] for i in self.list_of_points]

			plt.plot(
			    a_				,       a_				,       color='''blue'''				,       label='''Curve of Degree ''' + str(self.degree					)				,       )
			plt.scatter(a_				,       a_				,       color='''red'''				,       label='''Control Points'''					)
			plt.legend()
			plt.show()


if __name__ == "__main__":
			import doctest

			doctest.testmod()

			BezierCurve([(1, 2), (3, 5)]).plot_curve()  # degree 1
			BezierCurve([(0, 0), (5, 5), (5, 0)]).plot_curve()  # degree 2
			BezierCurve([(0, 0), (5, 5), (5, 0), (2.5, -2.5)]).plot_curve()  # degree 3





