Example 001: the number combination has four numbers: 1, 2, 3, 4. How many three digits can be formed that are different from each other and have no duplicate numbers? How many are each?

#  example 001： Number combination 
#  There are four numbers ：1、2、3、4, How many different and unrepeated three digit numbers can be formed ？ How many are each ？

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  First use mathematical permutation and combination analysis, and then Python Realization And summarize the methods

Catalog

         Analyze with mathematical permutation and combination

        Python Realization

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         Analyze with mathematical permutation and combination

        The first thing I think of is arranging the array composite , Because the title requires no repeated numbers , Therefore, it is the number of permutations . Definition of permutation ： from n Among the different elements , Take whatever you like m(m≤n,m And n All are natural numbers , The same below ） Different elements are arranged in a certain order , It's called from n Take out... Of the different elements m An arrangement of elements ; from n Take out... Of the different elements m(m≤n） The number of all permutations of elements , It's called from n Take out... Of the different elements m Number of permutations of elements .

                                       

          We can conclude that the permutation of this question is 4! / (4-3)! = 4! = 24 ( individual )

        Python Realization

         Method 1 ： We can loop through 3 A digital , Because it is required not to repeat numbers , Therefore, make conditional judgment when outputting , Only output hundreds of digits, tens of digits, and numbers with different single digits .

arrange = 0  #  Define the number of permutations 
nums = range(1, 5)

#  Method 1 
for i in nums:  #  Hundreds of digits 
    for j in nums:  #  Ten digits 
        for k in nums:  #  Single digit 
            if ((i != j) and (j != k) and (k != i)):  #  Only output numbers without repetition 
                print(i, j, k)
                arrange += 1
print(arrange)

The operation results are as follows ： 

1 2 3
1 2 4
1 3 2
1 3 4
1 4 2
1 4 3
2 1 3
2 1 4
2 3 1
2 3 4
2 4 1
2 4 3
3 1 2
3 1 4
3 2 1
3 2 4
3 4 1
3 4 2
4 1 2
4 1 3
4 2 1
4 2 3
4 3 1
4 3 2
24  individual 

Process finished with exit code 0

        Method 2 ： Can be called directly itertools Of permutations() class ,# itertools, yes python A built-in module for , Powerful , Mainly used for efficient loop creation iterators . Be careful. , He did not return list, It is iterator

import itertools  # itertools, yes python A built-in module for , Powerful , Mainly used for efficient loop creation iterators . Be careful. , He did not return list, It is iterator

arrange = 0  #  Define the number of permutations 
nums = range(1, 5)
for i in itertools.permutations(nums, 3):  # ( Iterating elements , Iteration sequence length )
    print(i)
    arrange += 1
print(arrange)

  The operation results are as follows ：

(1, 2, 3)
(1, 2, 4)
(1, 3, 2)
(1, 3, 4)
(1, 4, 2)
(1, 4, 3)
(2, 1, 3)
(2, 1, 4)
(2, 3, 1)
(2, 3, 4)
(2, 4, 1)
(2, 4, 3)
(3, 1, 2)
(3, 1, 4)
(3, 2, 1)
(3, 2, 4)
(3, 4, 1)
(3, 4, 2)
(4, 1, 2)
(4, 1, 3)
(4, 2, 1)
(4, 2, 3)
(4, 3, 1)
(4, 3, 2)
24

The second method is used itertools.permutations class , Such contents are as follows: ：

class permutations(object):
    """
    Return successive r-length permutations of elements in the iterable.
    
    permutations(range(3), 2) --> (0,1), (0,2), (1,0), (1,2), (2,0), (2,1)
    """
    def __getattribute__(self, *args, **kwargs): # real signature unknown
        """ Return getattr(self, name). """
        pass

    def __init__(self, range, *args, **kwargs): # real signature unknown; NOTE: unreliably restored from __doc__ 
        pass

    def __iter__(self, *args, **kwargs): # real signature unknown
        """ Implement iter(self). """
        pass

    @staticmethod # known case of __new__
    def __new__(*args, **kwargs): # real signature unknown
        """ Create and return a new object.  See help(type) for accurate signature. """
        pass

    def __next__(self, *args, **kwargs): # real signature unknown
        """ Implement next(self). """
        pass

    def __reduce__(self, *args, **kwargs): # real signature unknown
        """ Return state information for pickling. """
        pass

    def __setstate__(self, *args, **kwargs): # real signature unknown
        """ Set state information for unpickling. """
        pass

    def __sizeof__(self, *args, **kwargs): # real signature unknown
        """ Returns size in memory, in bytes. """
        pass

permutations Class to create an iterator , return iterable All lengths in are r Project sequence , If you omit r, Then the length of the sequence is the same as iterable The number of items in is the same , A full permutation result will be returned .