

from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
__A	:   int	=					20

# Time between time steps in seconds
__A	:   Any	=					INTERVAL / 10_00


class 			_SCREAMING_SNAKE_CASE      :







						def __init__(			self     ,						_SCREAMING_SNAKE_CASE     ,						_SCREAMING_SNAKE_CASE     ,						_SCREAMING_SNAKE_CASE     ,						_SCREAMING_SNAKE_CASE     ,						_SCREAMING_SNAKE_CASE = 1.0     ,						_SCREAMING_SNAKE_CASE = 1.0     ,						_SCREAMING_SNAKE_CASE = "blue"     ,						)->							None:
													lowerCamelCase_								=position_x
													lowerCamelCase_								=position_y
													lowerCamelCase_								=velocity_x
													lowerCamelCase_								=velocity_y
													lowerCamelCase_								=mass
													lowerCamelCase_								=size
													lowerCamelCase_								=color







						@property
						def        _snake_case				(			self							)->							tuple[float, float]:
													return self.position_x, self.position_y







						@property
						def        _snake_case				(			self							)->							tuple[float, float]:
													return self.velocity_x, self.velocity_y







						def        _snake_case				(			self     ,						_SCREAMING_SNAKE_CASE     ,						_SCREAMING_SNAKE_CASE     ,						_SCREAMING_SNAKE_CASE							)->							None:
													self.velocity_x += force_x * delta_time
													self.velocity_y += force_y * delta_time







						def        _snake_case				(			self     ,						_SCREAMING_SNAKE_CASE							)->							None:
													self.position_x += self.velocity_x * delta_time
													self.position_y += self.velocity_y * delta_time


class 			_SCREAMING_SNAKE_CASE      :







						def __init__(			self     ,						_SCREAMING_SNAKE_CASE     ,						_SCREAMING_SNAKE_CASE = 1.0     ,						_SCREAMING_SNAKE_CASE = 1.0     ,						_SCREAMING_SNAKE_CASE = 0.0     ,						)->							None:
													lowerCamelCase_								=bodies
													lowerCamelCase_								=gravitation_constant
													lowerCamelCase_								=time_factor
													lowerCamelCase_								=softening_factor







						def __len__(			self							)->							int:
													return len(self.bodies							)







						def        _snake_case				(			self     ,						_SCREAMING_SNAKE_CASE							)->							None:
													for bodya in self.bodies:
																				lowerCamelCase_								=0.0
																				lowerCamelCase_								=0.0
																				for bodya in self.bodies:
																											if bodya != bodya:
																																		lowerCamelCase_								=bodya.position_x - bodya.position_x
																																		lowerCamelCase_								=bodya.position_y - bodya.position_y

																																		# Calculation of the distance using Pythagoras's theorem
																																		# Extra factor due to the softening technique
																																		lowerCamelCase_								=(dif_x**2 + dif_y**2 + self.softening_factor) ** (
																																		    1 / 2
																																		)

																																		# Newton's law of universal gravitation.
																																		force_x += (
																																		    self.gravitation_constant * bodya.mass * dif_x / distance**3
																																		)
																																		force_y += (
																																		    self.gravitation_constant * bodya.mass * dif_y / distance**3
																																		)

            # Update the body's velocity once all the force components have been added
																				bodya.update_velocity(_SCREAMING_SNAKE_CASE     ,						_SCREAMING_SNAKE_CASE     ,						delta_time * self.time_factor							)

													# Update the positions only after all the velocities have been updated
													for body in self.bodies:
																				body.update_position(delta_time * self.time_factor							)




def        __UpperCamelCase     (      _A					:							BodySystem      ,  _A					:							float      ,  _A					:							list[plt.Circle]		)				->None:



							"""simple docstring"""
							body_system.update_system(_A		)

							# Update the positions of the patches
							for patch, body in zip(_A      ,  body_system.bodies		):
														lowerCamelCase_								=(body.position_x, body.position_y)




def        __UpperCamelCase     (      _A					:							str      ,  _A					:							BodySystem      ,  _A					:							float = -1      ,  _A					:							float = 1      ,  _A					:							float = -1      ,  _A					:							float = 1      ,  )				->None:



							"""simple docstring"""
							lowerCamelCase_								=plt.figure()
							fig.canvas.set_window_title(_A		)
							lowerCamelCase_								=plt.axes(
							    xlim=(x_start, x_end)      ,  ylim=(y_start, y_end)		)  # Set section to be plotted
							plt.gca().set_aspect("""equal"""		)  # Fix aspect ratio

							# Each body is drawn as a patch by the plt-function
							lowerCamelCase_								=[
							    plt.Circle((body.position_x, body.position_y)      ,  body.size      ,  fc=body.color		)
							    for body in body_system.bodies
							]

							for patch in patches:
														ax.add_patch(_A		)

							# Function called at each step of the animation
							def update(_A					:							int		) -> list[plt.Circle]:
														update_step(_A      ,  _A      ,  _A		)
														return patches

							lowerCamelCase_								=animation.FuncAnimation(  # noqa: F841
							    _A      ,  _A      ,  interval=_A      ,  blit=_A		)

							plt.show()




def        __UpperCamelCase     (      )				->BodySystem:



							"""simple docstring"""
							lowerCamelCase_								=0.9_7_0_0_4_3_6
							lowerCamelCase_								=-0.2_4_3_0_8_7_5_3
							lowerCamelCase_								=0.4_6_6_2_0_3_6_8_5
							lowerCamelCase_								=0.4_3_2_3_6_5_7_3

							lowerCamelCase_								=[
							    Body(_A      ,  _A      ,  _A      ,  _A      ,  size=0.2      ,  color="""red"""		),
							    Body(-position_x      ,  -position_y      ,  _A      ,  _A      ,  size=0.2      ,  color="""green"""		),
							    Body(0      ,  0      ,  -2 * velocity_x      ,  -2 * velocity_y      ,  size=0.2      ,  color="""blue"""		),
							]
							return BodySystem(_A      ,  time_factor=3		)




def        __UpperCamelCase     (      )				->BodySystem:



							"""simple docstring"""
							lowerCamelCase_								=7.3476E22
							lowerCamelCase_								=5.972E24
							lowerCamelCase_								=1022
							lowerCamelCase_								=384399000
							lowerCamelCase_								=6.674E-11

							# Calculation of the respective velocities so that total impulse is zero,
							# i.e. the two bodies together don't move
							lowerCamelCase_								=earth_mass * velocity_dif / (earth_mass + moon_mass)
							lowerCamelCase_								=moon_velocity - velocity_dif

							lowerCamelCase_								=Body(-earth_moon_distance      ,  0      ,  0      ,  _A      ,  _A      ,  10000000      ,  """grey"""		)
							lowerCamelCase_								=Body(0      ,  0      ,  0      ,  _A      ,  _A      ,  50000000      ,  """blue"""		)
							return BodySystem([earth, moon]      ,  _A      ,  time_factor=1000000		)




def        __UpperCamelCase     (      )				->BodySystem:



							"""simple docstring"""
							lowerCamelCase_								=[]
							for _ in range(10		):
														lowerCamelCase_								=random.uniform(-0.5      ,  0.5		)
														lowerCamelCase_								=random.uniform(-0.5      ,  0.5		)

														# Bodies are created pairwise with opposite velocities so that the
														# total impulse remains zero
														bodies.append(
														    Body(
														        random.uniform(-0.5      ,  0.5		)      ,  random.uniform(-0.5      ,  0.5		)      ,  _A      ,  _A      ,  size=0.0_5      ,  )		)
														bodies.append(
														    Body(
														        random.uniform(-0.5      ,  0.5		)      ,  random.uniform(-0.5      ,  0.5		)      ,  -velocity_x      ,  -velocity_y      ,  size=0.0_5      ,  )		)
							return BodySystem(_A      ,  0.0_1      ,  10      ,  0.1		)


if __name__ == "__main__":
	plot('Figure-8 solution to the 3-body-problem', example_a(), -2, 2, -2, 2)
	plot(
	    'Moon\'s orbit around the earth',
	    example_a(),
	    -4_30_00_00_00,
	    4_30_00_00_00,
	    -4_30_00_00_00,
	    4_30_00_00_00,
	)
	plot('Random system with many bodies', example_a(), -1.5, 1.5, -1.5, 1.5)



from math import factorial




def        __UpperCamelCase     (      _A					:							Union[str, Any]      ,  _A					:							Tuple		)				->Any:



							"""simple docstring"""
							return factorial(_A		) / (factorial(_A		) * factorial(n - r		))




def        __UpperCamelCase     (      )				->Any:



							"""simple docstring"""
							lowerCamelCase_								=0

							for i in range(1      ,  101		):
														for j in range(1      ,  i + 1		):
																					if combinations(_A      ,  _A		) > 1E6:
																												total += 1
							return total


if __name__ == "__main__":
	print(solution())
