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Nlogn

# DP - Coin Change: Find the number of ways of representing n cents

mayank joshi Originally published at nlogn.in ・2 min read

Given an infinite supply of 25 cents, 10 cents, 5 cents, and 1 cent. Find the number of ways of representing n cents. (The order doesn't matter).

``````Ex: n = 10 => {1,1,1,1,1,1,1,1,1,1}
{1,1,1,1,1,5}
{5,5}
{10}
``````

### Algorithm:

We are given a set of 4 Coins of type 1 cents, 5 cents, 10 cents, 25 cents. To find the number of ways of making n cents using these 4 cents, we will consider 2 conditions:

1. Try to make n cents by including the ith cent from the set of 4 coins.      `number_of_ways(n-coin_arr[m], coin_arr m);`   here m in the number of coins in given set(here m=4).
2. Try to make n cents by not including the ith cent from the given set of the 4 coins.  `number_of_ways(n, coin_arr, m-1);`

This problem involves the repetition of subproblems, so we will use Dynamic Programming.

`The number of ways of representing 6 cents using 1cents and 2 cents. (Here for simplicity we have considered a set of 2 coins only)`

### Implementation of the above code in CPP

1. Recursive Approach

2. Iterative Approach

### Output

``````Coins       Output
n = 5        2
n = 26       13
n = 1000     142511
``````

### Time Complexity

Since we have used Dynamic Programming here, hence the time complexity of the above program is O(NM), N the total cents to make and M is the number of the coin in a given set.

This post was originally published at nlogn