Find all combinations of a list of numbers with a given sum

Question

I have a list of numbers, e.g.

numbers = [1, 2, 3, 7, 7, 9, 10]

As you can see, numbers may appear more than once in this list.

I need to get all combinations of these numbers that have a given sum, e.g. 10.

The items in the combinations may not be repeated, but each item in numbers has to be treated uniquely, that means e.g. the two 7 in the list represent different items with the same value.

The order is unimportant, so that [1, 9] and [9, 1] are the same combination.

There are no length restrictions for the combinations, [10] is as valid as [1, 2, 7].

How can I create a list of all combinations meeting the criteria above?

In this example, it would be [[1,2,7], [1,2,7], [1,9], [3,7], [3,7], [10]]

Answer 1

You could use itertools to iterate through every combination of every possible size, and filter out everything that doesn't sum to 10:

import itertools

numbers = [1, 2, 3, 7, 7, 9, 10]
target = 10

result = [seq for i in range(len(numbers), 0, -1)
          for seq in itertools.combinations(numbers, i)
          if sum(seq) == target]

print(result)

Result:

[(1, 2, 7), (1, 2, 7), (1, 9), (3, 7), (3, 7), (10,)]

Unfortunately this is something like O(2^N) complexity, so it isn't suitable for input lists larger than, say, 20 elements.

Answer 2

The solution @kgoodrick offered is great but I think it is more useful as a generator:

def subset_sum(numbers, target, partial=[], partial_sum=0):
    if partial_sum == target:
        yield partial
    if partial_sum >= target:
        return
    for i, n in enumerate(numbers):
        remaining = numbers[i + 1:]
        yield from subset_sum(remaining, target, partial + [n], partial_sum + n)

Output:

print(list(subset_sum([1, 2, 3, 7, 7, 9, 10], 10)))
# [[1, 2, 7], [1, 2, 7], [1, 9], [3, 7], [3, 7], [10]]

Answer 3

This question has been asked before, see @msalvadores answer here. I updated the python code given to run in python 3:

def subset_sum(numbers, target, partial=[]):
    s = sum(partial)

    # check if the partial sum is equals to target
    if s == target:
        print("sum(%s)=%s" % (partial, target))
    if s >= target:
        return  # if we reach the number why bother to continue

    for i in range(len(numbers)):
        n = numbers[i]
        remaining = numbers[i + 1:]
        subset_sum(remaining, target, partial + [n])


if __name__ == "__main__":
    subset_sum([3, 3, 9, 8, 4, 5, 7, 10], 15)

    # Outputs:
    # sum([3, 8, 4])=15
    # sum([3, 5, 7])=15
    # sum([8, 7])=15
    # sum([5, 10])=15

Answer 4

@qasimalbaqali

This may not be exactly what the post is looking for, but if you wanted to:

Find all combinations of a range of numbers [lst], where each lst contains N number of elements, and that sum up to K: use this:

# Python3 program to find all pairs in a list of integers with given sum  
from itertools import combinations 

def findPairs(lst, K, N): 
    return [pair for pair in combinations(lst, N) if sum(pair) == K] 

#monthly cost range; unique numbers
lst = list(range(10, 30))
#sum of annual revenue per machine/customer
K = 200
#number of months (12 - 9 = num months free)
N = 9

print('Possible monthly subscription costs that still equate to $200 per year:')
#print(findPairs(lst, K, N)) 
findPairs(lst,K,N)

Results:

Possible monthly subscription costs that still equate to $200 per year:
Out[27]:
[(10, 11, 20, 24, 25, 26, 27, 28, 29),
 (10, 11, 21, 23, 25, 26, 27, 28, 29),
 (10, 11, 22, 23, 24, 26, 27, 28, 29),

The idea/question behind this was "how much can we charge per month if we give x number of months free and still meet revenue targets".

Answer 5

This works...

from itertools import combinations


def SumTheList(thelist, target):
    arr = []
    p = []    
    if len(thelist) > 0:
        for r in range(0,len(thelist)+1):        
            arr += list(combinations(thelist, r))

        for item in arr:        
            if sum(item) == target:
                p.append(item)

    return p

Answer 6

Append: including zero.

import random as rd

def combine(soma, menor, maior):
    """All combinations of 'n' sticks and '3' plus sinals.
    seq = [menor, menor+1, ..., maior]
    menor = min(seq); maior = max(seq)"""
    lista = []

    while len(set(lista)) < 286: # This number is defined by the combination
                                 # of (sum + #summands - 1, #summands - 1) -> choose(13, 3)     
        zero = rd.randint(menor, maior)

        if zero == soma and (zero, 0, 0, 0) not in lista:
            lista.append((zero, 0, 0, 0))

        else:
            # You can add more summands!

            um = rd.randint(0, soma - zero)
            dois = rd.randint(0, soma - zero - um)
            tres = rd.randint(0, soma - zero - um - dois)


            if (zero + um + dois + tres  == soma and
             (zero, um, dois, tres) not in lista):
                lista.append((zero, um, dois, tres))

    return sorted(lista)

>>> result_sum = 10
>>> combine(result_sum, 0, 10)

Output

[(0,0,0,10), (0,0,1,9), (0,0,2,8), (0,0,3,7), ...,
(9,1,0,0), (10,0,0,0)]