Source

Part 1: Given a list of containers of varying sizes and a total volume to contain, determine how many different combinations of containers match exactly the given volume.

containers = list(map(int, sys.stdin.readlines()))
quantity = int(sys.argv[1])

def fills(quantity, containers):
    if quantity == 0:
        yield []
    else:
        for index, container in enumerate(containers):
            if container <= quantity:
                for sub_fill in fills(quantity - container, containers[index+1:]):
                    yield [container] + sub_fill

print(len(list(fills(quantity, containers))))

Basically, fills will recursively build a generator for a given amount and containers remaining. For each step, it will try each of the containers and recursively determine any solutions that use that container.

On interesting aspect is the recursive call uses containers[index+1:]. That will guarantee that the containers used are always in the same order as the input list, rather than including all possible orderings for any set of containers.

Part 2: How many of these combinations use the minimum number of containers?

The fills function still does what we need; we just need to filter it so that it only includes orderings that are the proper minimum length:

smallest_fill = min(map(len, fills(quantity, containers)))

print(len(list(filter(
    lambda fill : len(fill) == smallest_fill,
    fills(quantity, containers)
))))