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from sympy.core.numbers import Rational
from sympy.core.symbol import symbols
from sympy.functions.combinatorial.factorials import (FallingFactorial, RisingFactorial, binomial, factorial)
from sympy.functions.special.gamma_functions import gamma
from sympy.simplify.combsimp import combsimp
from sympy.abc import x
def test_combsimp():
k, m, n = symbols('k m n', integer = True)
assert combsimp(factorial(n)) == factorial(n)
assert combsimp(binomial(n, k)) == binomial(n, k)
assert combsimp(factorial(n)/factorial(n - 3)) == n*(-1 + n)*(-2 + n)
assert combsimp(binomial(n + 1, k + 1)/binomial(n, k)) == (1 + n)/(1 + k)
assert combsimp(binomial(3*n + 4, n + 1)/binomial(3*n + 1, n)) == \
Rational(3, 2)*((3*n + 2)*(3*n + 4)/((n + 1)*(2*n + 3)))
assert combsimp(factorial(n)**2/factorial(n - 3)) == \
factorial(n)*n*(-1 + n)*(-2 + n)
assert combsimp(factorial(n)*binomial(n + 1, k + 1)/binomial(n, k)) == \
factorial(n + 1)/(1 + k)
assert combsimp(gamma(n + 3)) == factorial(n + 2)
assert combsimp(factorial(x)) == gamma(x + 1)
# issue 9699
assert combsimp((n + 1)*factorial(n)) == factorial(n + 1)
assert combsimp(factorial(n)/n) == factorial(n-1)
# issue 6658
assert combsimp(binomial(n, n - k)) == binomial(n, k)
# issue 6341, 7135
assert combsimp(factorial(n)/(factorial(k)*factorial(n - k))) == \
binomial(n, k)
assert combsimp(factorial(k)*factorial(n - k)/factorial(n)) == \
1/binomial(n, k)
assert combsimp(factorial(2*n)/factorial(n)**2) == binomial(2*n, n)
assert combsimp(factorial(2*n)*factorial(k)*factorial(n - k)/
factorial(n)**3) == binomial(2*n, n)/binomial(n, k)
assert combsimp(factorial(n*(1 + n) - n**2 - n)) == 1
assert combsimp(6*FallingFactorial(-4, n)/factorial(n)) == \
(-1)**n*(n + 1)*(n + 2)*(n + 3)
assert combsimp(6*FallingFactorial(-4, n - 1)/factorial(n - 1)) == \
(-1)**(n - 1)*n*(n + 1)*(n + 2)
assert combsimp(6*FallingFactorial(-4, n - 3)/factorial(n - 3)) == \
(-1)**(n - 3)*n*(n - 1)*(n - 2)
assert combsimp(6*FallingFactorial(-4, -n - 1)/factorial(-n - 1)) == \
-(-1)**(-n - 1)*n*(n - 1)*(n - 2)
assert combsimp(6*RisingFactorial(4, n)/factorial(n)) == \
(n + 1)*(n + 2)*(n + 3)
assert combsimp(6*RisingFactorial(4, n - 1)/factorial(n - 1)) == \
n*(n + 1)*(n + 2)
assert combsimp(6*RisingFactorial(4, n - 3)/factorial(n - 3)) == \
n*(n - 1)*(n - 2)
assert combsimp(6*RisingFactorial(4, -n - 1)/factorial(-n - 1)) == \
-n*(n - 1)*(n - 2)
def test_issue_6878():
n = symbols('n', integer=True)
assert combsimp(RisingFactorial(-10, n)) == 3628800*(-1)**n/factorial(10 - n)
def test_issue_14528():
p = symbols("p", integer=True, positive=True)
assert combsimp(binomial(1,p)) == 1/(factorial(p)*factorial(1-p))
assert combsimp(factorial(2-p)) == factorial(2-p)
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