### Python's combinations and combinations with replacement

Author: Ahmedur Rahman Shovon

### Python’s combinations and combinations with replacement

A recent Stack Overflow question had me looking at how to find a list of four numbers each from -1 to 1 that sum up to 1.

### Problem statement

Generate every combination of four numbers(a.k.a. quadruplets) from `(-1, -0.99, -0.98, ..., 0.98, 0.99, 1)`

that sum up to 1. Repetitions of numbers are allowed.
For example:
`[0, 1, 0.12, -0.12], [0, 0, 0.5, 0.5], [1, 1, -1, 0]...`

.

### Solution using Python combination

I assume there will be only a set of unique lists that we should compute. This assumption drives me to use Python’s combination method to get a list of quadruplets from a list of numbers from `-1.0`

to `+1.0`

which sum up to 1.

Operating floating numbers are complex, so I multiplied the target sum with 100 and made integer calculations. I created a list from `-100`

to `+100`

and repeated the elements four times as the problem allowed repetitions of numbers. Later on, I made a list of quadruplets which sum is equal to 100. Finally, I divided each element by 100 to meet the problem statement.

So far, the code looks like the following snippet:

```
import csv
from itertools import combinations
def get_four_numbers(target_sum=100):
result = []
ar = list(range(-100, 101, 1)) * 4
for quadruplet in combinations(ar, 4):
if sum(quadruplet) == target_sum:
result.append([i / 100.0 for i in quadruplet])
return result
if __name__ == "__main__":
with open("output.csv", "w") as output_file:
csv_writer = csv.writer(output_file)
csv_writer.writerows(get_four_numbers())
```

### Solution using Python combination

Later on, I found another method from the `itertools`

module called combination with replacement. This will allow me to get the targeted quadruplets without repeating the iterable list `ar`

four times manually. Updating the above snippet, we now have a much cleaner code with the same output quadruplets:

```
import csv
from itertools import combinations_with_replacement
def get_four_numbers(target_sum=100):
result = []
ar = list(range(-100, 101, 1))
for quadruplet in combinations_with_replacement(ar, 4):
if sum(quadruplet) == target_sum:
result.append([i / 100.0 for i in quadruplet])
return result
if __name__ == "__main__":
with open("output.csv", "w") as output_file:
csv_writer = csv.writer(output_file)
csv_writer.writerows(get_four_numbers())
```

**Output from output.csv (truncated due to filesize):**

```
-1.0,0.0,1.0,1.0
-1.0,0.01,0.99,1.0
-1.0,0.02,0.98,1.0
-1.0,0.02,0.99,0.99
-1.0,0.03,0.97,1.0
-1.0,0.03,0.98,0.99
-1.0,0.04,0.96,1.0
.....
```

### Reference

- Python official documentation on combination method
- Python official documentation on combination with replacement method
- Stackoverflow question: Four numbers each from -1 to 1 that sum up to 1

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