Q:

Coin Change

Given a value N, find the number of ways to make change for N cents, if we have infinite supply of each of S = { S1, S2, .. , Sm} valued coins. The order of coins doesn't matter.

Example:

```    Input:
N = 4
S = {1, 2, 3} //infinite number of 1 cent, 2 cent, 3 cent coins

Output:
4
{1,1,1,1} //four 1 cent coins
{1,1,2} //two 1 cent coins, one 2 cent coins
{2,2} //two 2 cent coins
{1,3} //one 1 cent and one 3 cent coins
Thus total four ways (Order doesn’t matter)```

Let's think of the solution. Let’s do it by hand first. An intuitive idea can be two checks whether the amount can be handled by the same valued coins.

Like 4 cent can be managed by four 1 cent coins.

Same time can be managed by two 2 cent coins.

For the rest, we need to take different valued coins.

Now we can think of memorization. We can simply store the sub-amounts that can have been managed still.

Like two 1 cent coin manage 2 cents. Thus 2 cent is our sub amount.

Now we pick two more 1 cent coins that will sum up to 4.

Else we can pick only one 2cent coin also summing up to 4.

This memorized approach helps us to solve the problem.

Let's revise the algorithm

Pre-requisite: Coins array, amount

Algorithm:

```    1.  Create a DP table of size amount
Table[amount+1]={0};
2.  Base case table[0]=1
3.  For each coins[i] from the coins array
For j= 1: amount //j be the sub-amount
IF j>=coins[i]
Table[j]=table[j]+table[j-coins[i]]
END IF
END For
END For
4.  Return table[amount]
Table[amount] refers to all possible way to make the amount
```

Example with explanation:

```    We have pre added coin 0-cent as coin[0]=0
For Coin index:  1 //coins[1]=1

Sub amount:1

Table status:
1 | 1 | 0 | 0 | 0

Sub amount:2
Table status:
1 | 1 | 1 | 0 | 0

Sub amount:3

1 | 1 | 1 | 1 | 0
Sub amount:4
1 | 1 | 1 | 1 | 1
```

This actually shows that by coin[1] all the sub-amounts can be managed only by one way, so if we continue this way for other coins we will get all the possible ways.

C++ implementation:

``````#include <bits/stdc++.h>
using namespace std;

void findway(vector<int> a,int n,int amount){
int table[amount+1]; //DP table
memset(table,0,sizeof(table));
table[0]=1;
//j be the sub-amounts
for(int i=1;i<=n;i++){
for(int j=1;j<=amount;j++){
if(j>=a[i])
table[j]+=table[j-a[i]];
}
}
cout<<table[amount]<<endl; //final result
}

int main()
{
int n,item,amount;

cout<<"Enter the number of coins\n";
scanf("%d",&n);

cout<<"Enter value of coins\n";
vector<int> a; //coins array
//we pre add 0-cent coin as coins[0]
//for sake of 1 indexing
a.push_back(0);

for(int j=0;j<n;j++){
scanf("%d",&item);
a.push_back(item);
}

cout<<"Enter total amount\n";
cin>>amount;
cout<<"Number of ways to sum the amount is: ";
findway(a,n,amount);

return 0;
}``````

Output

```Enter the number of coins
3
Enter value of coins
1 2 3
Enter total amount
4
Number of ways to sum the amount is: 4```