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Mathematics

Finding nCr for given numbers​

import math

def nCr(n,r):
        f = math.factorial
    return f(n) // f(r) // f(n-r)

Use memoization if large number of calls is to be made

Permutation​

First import itertools package to implement permutations method in python. This method takes a list as an input and return an object list of tuples that contain all permutation in a list form.

# A Python program to print all
# permutations using library function
from itertools import permutations

# Get all permutations of [1, 2, 3]
perm = permutations([1, 2, 3])

# Print the obtained permutations
for i in list(perm):
    print i

Output
(1, 2, 3)
(1, 3, 2)
(2, 1, 3)
(2, 3, 1)
(3, 1, 2)
(3, 2, 1)

It generates n! permutations if length of input sequence is n. If want want to get permutations of length L then implement it in this way.

# A Python program to print all
# permutations of given length
from itertools import permutations

# Get all permutations of length 2
# and length 2
perm = permutations([1, 2, 3], 2)

# Print the obtained permutations
for i in list(perm):
    print i

Output
(1, 2)
(1, 3)
(2, 1)
(2, 3)
(3, 1)
(3, 2)

It generate nCr * r! permutations if length of input sequence is n and input parameter is r.

Combination​

This method takes a list and r (length) as an input and return an object list of tuples which contain all possible combination of length r in a list form.

# A Python program to print all
# combinations of given length
from itertools import combinations

# Get all combinations of [1, 2, 3]
# and length 2
comb = combinations([1, 2, 3], 2)

# Print the obtained combinations
for i in list(comb):
    print i

Output
(1, 2)
(1, 3)
(2, 3)

1. Combinations are emitted in lexicographic sort order of input. So, if the input list is sorted, the combination tuples will be produced in sorted order​

# A Python program to print all combinations
# of given length with unsorted input.
from itertools import combinations

# Get all combinations of [2, 1, 3]
# and length 2
comb = combinations([2, 1, 3], 2)

# Print the obtained combinations
for i in list(comb):
    print i

Output
(2, 1)
(2, 3)
(1, 3)

2. Elements are treated as unique based on their position, not on their value. So if the input elements are unique, there will be no repeat values in each combination​

# A Python program to print all combinations
# of given length with duplicates in input
from itertools import combinations

# Get all combinations of [1, 1, 3]
# and length 2
comb = combinations([1, 1, 3], 2)

# Print the obtained combinations
for i in list(comb):
    print i

Output
(1, 1)
(1, 3)
(1, 3)

3. If we want to make combination of same element to same element then we use combinations_with_replacement​

# A Python program to print all combinations
# with an element-to-itself combination is
# also included
from itertools import combinations_with_replacement

# Get all combinations of [1, 2, 3] and length 2
comb = combinations_with_replacement([1, 2, 3], 2)

# Print the obtained combinations
for i in list(comb):
    print i

Output
(1, 1)
(1, 2)
(1, 3)
(2, 2)
(2, 3)
(3, 3)