#finished version
def problem31(target, coins):
    ways = [0] * (target + 1)
    ways[0] = 1
    for i in range(len(coins)):
        for j in range(coins[i], target + 1, 1):
            ways[j] += ways[j - coins[i]]
    return ways[target]


#easier to follow
def problem31(target, coins):
    ways = [0] * (target + 1)
    ways[0] = 1
    for i in range(len(coins)):
        print("coin value:", coins[i])
        for j in range(coins[i], target + 1, 1):
            print(ways[j], "+", ways[j - coins[i]], end="\t\t")
            ways[j] += ways[j - coins[i]]
            print("ways[%s] :" % j, ways[j])
    return ways[target]


example way of running it:
problem31(200, [1, 2, 5, 10, 20, 50, 100, 200])