Combinatoric Functions

'''
combonatorics 4/12/2026
Edward Shore

round to nearest integer:
int(f+.5)
'''

from math import *

# factorial positive integers
def fact(n):
  f,i=1,1
  while i<=n:
    f*=i
    i+=1
  return int(f+.5)

# a lot of functions will use fact

# combination
def ncr(n,r):
  f=fact(n)/(fact(r)*fact(n-r))
  return int(f+.5)

# permutation
def npr(n,r):
  f=fact(n)/fact(n-r)
  return int(f+.5)

# combination w/repetitions
def nhr(n,r):
  f=fact(n+r-1)/(fact(r)*fact(n-1))
  return int(f+.5)

# catalan numbers
def catalan(n):
  f=fact(2*n)/(fact(n)**2*(n+1))
  return int(f+.5)

# narayana numbers
def narayana(n,k):
  f=1/n*ncr(n,k)*ncr(n,k-1)
  return int(f+.5)

# binomial probability
def binpdf(n,k,p):
  # p=prob
  f=ncr(n,k)*p**k*(1-p)**(n-k)
  return f

# bin lower tail
def bincdf(n,k,p):
  # 0 to k, p=prob
  s=0
  for i in range(k+1):
    s+=binpdf(n,i,p)
  return s