Problem Description

In England the currency is made up of pound, £, and pence, p, and there are eight coins in general circulation:

1p, 2p, 5p, 10p, 20p, 50p, £1 (100p) and £2 (200p).
It is possible to make £2 in the following way:

1×£1 + 1×50p + 2×20p + 1×5p + 1×2p + 3×1p
How many different ways can £2 be made using any number of coins?

Analysis

This is a typical problem that demonstrates the use of partitions which can be solved by using dynamic programming. To keep this problem simple: order does not matter, there are always enough coins to make a combination and we’re not looking for the optimal way to make change. This a perfect application for tested algorithms where this problem has been used as an example. As the details and nuances of dynamic programming are somewhat involved; we have provided some links below for a better explanation:
Coin Change – Algorithmist
Dynamic Programming Solution to the Coin Changing Problem

Solution

Runs < 1 second in Python.

target = 200
coins = [1,2,5,10,20,50,100,200]
ways = [1]+[0]*target
 
for coin in coins:
  for i in range(coin, target+1):
    ways[i] += ways[i-coin]
 
print "Answer to PE31 = ", ways[target]

Comments

• See Problem 76. This problem only wants the number of combinations. It would be a different approach if they wanted a set of combinations.