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Coin Change | DP-7
• Difficulty Level : Hard
• Last Updated : 03 May, 2021

Given a value N, if we want to make change for N cents, and we have infinite supply of each of S = { S1, S2, .. , Sm} valued coins, how many ways can we make the change? The order of coins doesn’t matter.
For example, for N = 4 and S = {1,2,3}, there are four solutions: {1,1,1,1},{1,1,2},{2,2},{1,3}. So output should be 4. For N = 10 and S = {2, 5, 3, 6}, there are five solutions: {2,2,2,2,2}, {2,2,3,3}, {2,2,6}, {2,3,5} and {5,5}. So the output should be 5.

1) Optimal Substructure
To count the total number of solutions, we can divide all set solutions into two sets.
1) Solutions that do not contain mth coin (or Sm).
2) Solutions that contain at least one Sm.
Let count(S[], m, n) be the function to count the number of solutions, then it can be written as sum of count(S[], m-1, n) and count(S[], m, n-Sm).
Therefore, the problem has optimal substructure property as the problem can be solved using solutions to subproblems.
2) Overlapping Subproblems
Following is a simple recursive implementation of the Coin Change problem. The implementation simply follows the recursive structure mentioned above.

## C++

 `// Recursive C program for``// coin change problem.``#include` `// Returns the count of ways we can``// sum S[0...m-1] coins to get sum n``int` `count( ``int` `S[], ``int` `m, ``int` `n )``{``    ``// If n is 0 then there is 1 solution``    ``// (do not include any coin)``    ``if` `(n == 0)``        ``return` `1;``    ` `    ``// If n is less than 0 then no``    ``// solution exists``    ``if` `(n < 0)``        ``return` `0;` `    ``// If there are no coins and n``    ``// is greater than 0, then no``    ``// solution exist``    ``if` `(m <=0 && n >= 1)``        ``return` `0;` `    ``// count is sum of solutions (i)``    ``// including S[m-1] (ii) excluding S[m-1]``    ``return` `count( S, m - 1, n ) + count( S, m, n-S[m-1] );``}` `// Driver program to test above function``int` `main()``{``    ``int` `i, j;``    ``int` `arr[] = {1, 2, 3};``    ``int` `m = ``sizeof``(arr)/``sizeof``(arr);``    ``printf``(``"%d "``, count(arr, m, 4));``    ``getchar``();``    ``return` `0;``}`

Java

``````
// Recursive java program for
// coin change problem.
import java.io.*;

class GFG {

// Returns the count of ways we can
// sum S[0...m-1] coins to get sum n
static int count( int S[], int m, int n )
{
// If n is 0 then there is 1 solution
// (do not include any coin)
if (n == 0)
return 1;

// If n is less than 0 then no
// solution exists
if (n < 0)
return 0;

// If there are no coins and n
// is greater than 0, then no
// solution exist
if (m <=0 && n >= 1)
return 0;

// count is sum of solutions (i)
// including S[m-1] (ii) excluding S[m-1]
return count( S, m - 1, n ) +
count( S, m, n-S[m-1] );
}

// Driver program to test above function
public static void main(String[] args)
{
int arr[] = {1, 2, 3};
int m = arr.length;
System.out.println( count(arr, m, 4));

}

}

``````

## Python3

 `# Recursive Python3 program for``# coin change problem.` `# Returns the count of ways we can sum``# S[0...m-1] coins to get sum n``def` `count(S, m, n ):` `    ``# If n is 0 then there is 1``    ``# solution (do not include any coin)``    ``if` `(n ``=``=` `0``):``        ``return` `1` `    ``# If n is less than 0 then no``    ``# solution exists``    ``if` `(n < ``0``):``        ``return` `0``;` `    ``# If there are no coins and n``    ``# is greater than 0, then no``    ``# solution exist``    ``if` `(m <``=``0` `and` `n >``=` `1``):``        ``return` `0` `    ``# count is sum of solutions (i)``    ``# including S[m-1] (ii) excluding S[m-1]``    ``return` `count( S, m ``-` `1``, n ) ``+` `count( S, m, n``-``S[m``-``1``] );` `# Driver program to test above function``arr ``=` `[``1``, ``2``, ``3``]``m ``=` `len``(arr)``print``(count(arr, m, ``4``))` `# This code is contributed by Smitha Dinesh Semwal`

## C#

 `// Recursive C# program for``// coin change problem.``using` `System;` `class` `GFG``{``    ``// Returns the count of ways we can``    ``// sum S[0...m-1] coins to get sum n``    ``static` `int` `count( ``int` `[]S, ``int` `m, ``int` `n )``    ``{``        ``// If n is 0 then there is 1 solution``        ``// (do not include any coin)``        ``if` `(n == 0)``            ``return` `1;``        ` `        ``// If n is less than 0 then no``        ``// solution exists``        ``if` `(n < 0)``            ``return` `0;``    ` `        ``// If there are no coins and n``        ``// is greater than 0, then no``        ``// solution exist``        ``if` `(m <=0 && n >= 1)``            ``return` `0;``    ` `        ``// count is sum of solutions (i)``        ``// including S[m-1] (ii) excluding S[m-1]``        ``return` `count( S, m - 1, n ) +``            ``count( S, m, n - S[m - 1] );``    ``}``    ` `    ``// Driver program``    ``public` `static` `void` `Main()``    ``{``        ` `        ``int` `[]arr = {1, 2, 3};``        ``int` `m = arr.Length;``        ``Console.Write( count(arr, m, 4));``        ` `        ` `    ``}``}``// This code is contributed by Sam007`

## PHP

 `= 1)``        ``return` `0;` `    ``// count is sum of solutions (i)``    ``// including S[m-1] (ii) excluding S[m-1]``    ``return` `coun(``\$S``, ``\$m` `- 1,``\$n` `) +``           ``coun(``\$S``, ``\$m``, ``\$n` `- ``\$S``[``\$m` `- 1] );``}` `    ``// Driver Code``    ``\$arr` `= ``array``(1, 2, 3);``    ``\$m` `= ``count``(``\$arr``);``    ``echo` `coun(``\$arr``, ``\$m``, 4);``    ` `// This code is contributed by Sam007``?>`

## Javascript

 ``

Output :

`4`

It should be noted that the above function computes the same subproblems again and again. See the following recursion tree for S = {1, 2, 3} and n = 5.
The function C({1}, 3) is called two times. If we draw the complete tree, then we can see that there are many subproblems being called more than once.

```C() --> count()
C({1,2,3}, 5)
/             \
/                 \
C({1,2,3}, 2)                 C({1,2}, 5)
/       \                      /      \
/         \                    /         \
C({1,2,3}, -1)  C({1,2}, 2)        C({1,2}, 3)    C({1}, 5)
/    \             /     \           /     \
/       \           /       \         /        \
C({1,2},0)  C({1},2)   C({1,2},1) C({1},3)    C({1}, 4)  C({}, 5)
/ \     / \        /\         /     \
/   \   /   \     /   \       /       \
.      .  .     .   .     .   C({1}, 3) C({}, 4)
/ \
/   \
.      .```

Since same suproblems are called again, this problem has Overlapping Subprolems property. So the Coin Change problem has both properties (see this and this) of a dynamic programming problem. Like other typical Dynamic Programming(DP) problems, recomputations of same subproblems can be avoided by constructing a temporary array table[][] in bottom up manner.
Dynamic Programming Solution

## C++

 `// C++ program for coin change problem.``#include` `using` `namespace` `std;` `int` `count( ``int` `S[], ``int` `m, ``int` `n )``{``    ``int` `i, j, x, y;` `    ``// We need n+1 rows as the table``    ``// is constructed in bottom up``    ``// manner using the base case 0``    ``// value case (n = 0)``    ``int` `table[n + 1][m];``    ` `    ``// Fill the enteries for 0``    ``// value case (n = 0)``    ``for` `(i = 0; i < m; i++)``        ``table[i] = 1;` `    ``// Fill rest of the table entries``    ``// in bottom up manner``    ``for` `(i = 1; i < n + 1; i++)``    ``{``        ``for` `(j = 0; j < m; j++)``        ``{``            ``// Count of solutions including S[j]``            ``x = (i-S[j] >= 0) ? table[i - S[j]][j] : 0;` `            ``// Count of solutions excluding S[j]``            ``y = (j >= 1) ? table[i][j - 1] : 0;` `            ``// total count``            ``table[i][j] = x + y;``        ``}``    ``}``    ``return` `table[n][m - 1];``}` `// Driver Code``int` `main()``{``    ``int` `arr[] = {1, 2, 3};``    ``int` `m = ``sizeof``(arr)/``sizeof``(arr);``    ``int` `n = 4;``    ``cout << count(arr, m, n);``    ``return` `0;``}` `// This code is contributed``// by Akanksha Rai(Abby_akku)`

## C

 `// C program for coin change problem.``#include` `int` `count( ``int` `S[], ``int` `m, ``int` `n )``{``    ``int` `i, j, x, y;` `    ``// We need n+1 rows as the table is constructed``    ``// in bottom up manner using the base case 0``    ``// value case (n = 0)``    ``int` `table[n+1][m];``   ` `    ``// Fill the enteries for 0 value case (n = 0)``    ``for` `(i=0; i= 0)? table[i - S[j]][j]: 0;` `            ``// Count of solutions excluding S[j]``            ``y = (j >= 1)? table[i][j-1]: 0;` `            ``// total count``            ``table[i][j] = x + y;``        ``}``    ``}``    ``return` `table[n][m-1];``}` `// Driver program to test above function``int` `main()``{``    ``int` `arr[] = {1, 2, 3};``    ``int` `m = ``sizeof``(arr)/``sizeof``(arr);``    ``int` `n = 4;``    ``printf``(``" %d "``, count(arr, m, n));``    ``return` `0;``}`

## Java

 `/* Dynamic Programming Java implementation of Coin``   ``Change problem */``import` `java.util.Arrays;` `class` `CoinChange``{``    ``static` `long` `countWays(``int` `S[], ``int` `m, ``int` `n)``    ``{``        ``//Time complexity of this function: O(mn)``        ``//Space Complexity of this function: O(n)` `        ``// table[i] will be storing the number of solutions``        ``// for value i. We need n+1 rows as the table is``        ``// constructed in bottom up manner using the base``        ``// case (n = 0)``        ``long``[] table = ``new` `long``[n+``1``];` `        ``// Initialize all table values as 0``        ``Arrays.fill(table, ``0``);   ``//O(n)` `        ``// Base case (If given value is 0)``        ``table[``0``] = ``1``;` `        ``// Pick all coins one by one and update the table[]``        ``// values after the index greater than or equal to``        ``// the value of the picked coin``        ``for` `(``int` `i=``0``; i

## Python

 `# Dynamic Programming Python implementation of Coin``# Change problem``def` `count(S, m, n):``    ``# We need n+1 rows as the table is constructed``    ``# in bottom up manner using the base case 0 value``    ``# case (n = 0)``    ``table ``=` `[[``0` `for` `x ``in` `range``(m)] ``for` `x ``in` `range``(n``+``1``)]` `    ``# Fill the entries for 0 value case (n = 0)``    ``for` `i ``in` `range``(m):``        ``table[``0``][i] ``=` `1` `    ``# Fill rest of the table entries in bottom up manner``    ``for` `i ``in` `range``(``1``, n``+``1``):``        ``for` `j ``in` `range``(m):` `            ``# Count of solutions including S[j]``            ``x ``=` `table[i ``-` `S[j]][j] ``if` `i``-``S[j] >``=` `0` `else` `0` `            ``# Count of solutions excluding S[j]``            ``y ``=` `table[i][j``-``1``] ``if` `j >``=` `1` `else` `0` `            ``# total count``            ``table[i][j] ``=` `x ``+` `y` `    ``return` `table[n][m``-``1``]` `# Driver program to test above function``arr ``=` `[``1``, ``2``, ``3``]``m ``=` `len``(arr)``n ``=` `4``print``(count(arr, m, n))` `# This code is contributed by Bhavya Jain`

## C#

 `/* Dynamic Programming C# implementation of Coin``Change problem */``using` `System;` `class` `GFG``{``    ``static` `long` `countWays(``int` `[]S, ``int` `m, ``int` `n)``    ``{``        ``//Time complexity of this function: O(mn)``        ``//Space Complexity of this function: O(n)` `        ``// table[i] will be storing the number of solutions``        ``// for value i. We need n+1 rows as the table is``        ``// constructed in bottom up manner using the base``        ``// case (n = 0)``        ``int``[] table = ``new` `int``[n+1];` `        ``// Initialize all table values as 0``        ``for``(``int` `i = 0; i < table.Length; i++)``        ``{``            ``table[i] = 0;``        ``}` `        ``// Base case (If given value is 0)``        ``table = 1;` `        ``// Pick all coins one by one and update the table[]``        ``// values after the index greater than or equal to``        ``// the value of the picked coin``        ``for` `(``int` `i = 0; i < m; i++)``            ``for` `(``int` `j = S[i]; j <= n; j++)``                ``table[j] += table[j - S[i]];` `        ``return` `table[n];``    ``}` `    ``// Driver Function``    ``public` `static` `void` `Main()``    ``{``        ``int` `[]arr = {1, 2, 3};``        ``int` `m = arr.Length;``        ``int` `n = 4;``        ``Console.Write(countWays(arr, m, n));``    ``}``}``// This code is contributed by Sam007`

## PHP

 `= 0) ?``                  ``\$table``[``\$i` `- ``\$S``[``\$j``]][``\$j``] : 0;` `            ``// Count of solutions``            ``// excluding S[j]``            ``\$y` `= (``\$j` `>= 1) ?``                  ``\$table``[``\$i``][``\$j` `- 1] : 0;` `            ``// total count``            ``\$table``[``\$i``][``\$j``] = ``\$x` `+ ``\$y``;``        ``}``    ``}``    ``return` `\$table``[``\$n``][``\$m``-1];``}` `// Driver Code``\$arr` `= ``array``(1, 2, 3);``\$m` `= ``count``(``\$arr``);``\$n` `= 4;``echo` `count1(``\$arr``, ``\$m``, ``\$n``);` `// This code is contributed by mits``?>`

## Javascript

 ``

Output:

`4`

Time Complexity: O(mn)
Following is a simplified version of method 2. The auxiliary space required here is O(n) only.

## C++

 `int` `count( ``int` `S[], ``int` `m, ``int` `n )         ``{         `` ``// table[i] will be storing the number of solutions for         `` ``// value i. We need n+1 rows as the table is constructed         `` ``// in bottom up manner using the base case (n = 0)         `` ``int` `table[n+1];         `` ``// Initialize all table values as 0         `` ``memset``(table, 0, ``sizeof``(table));         `` ``// Base case (If given value is 0)         `` ``table = 1;         `` ``// Pick all coins one by one and update the table[] values         `` ``// after the index greater than or equal to the value of the         `` ``// picked coin         `` ``for``(``int` `i=0; i

## Java

 `public` `static` `int` `count( ``int` `S[], ``int` `m, ``int` `n )``{``    ``// table[i] will be storing the number of solutions for``    ``// value i. We need n+1 rows as the table is constructed``    ``// in bottom up manner using the base case (n = 0)``    ``int` `table[]=``new` `int``[n+``1``];` `    ``// Base case (If given value is 0)``    ``table[``0``] = ``1``;` `    ``// Pick all coins one by one and update the table[] values``    ``// after the index greater than or equal to the value of the``    ``// picked coin``    ``for``(``int` `i=``0``; i

## Python

 `# Dynamic Programming Python implementation of Coin``# Change problem``def` `count(S, m, n):` `    ``# table[i] will be storing the number of solutions for``    ``# value i. We need n+1 rows as the table is constructed``    ``# in bottom up manner using the base case (n = 0)``    ``# Initialize all table values as 0``    ``table ``=` `[``0` `for` `k ``in` `range``(n``+``1``)]` `    ``# Base case (If given value is 0)``    ``table[``0``] ``=` `1` `    ``# Pick all coins one by one and update the table[] values``    ``# after the index greater than or equal to the value of the``    ``# picked coin``    ``for` `i ``in` `range``(``0``,m):``        ``for` `j ``in` `range``(S[i],n``+``1``):``            ``table[j] ``+``=` `table[j``-``S[i]]` `    ``return` `table[n]` `# Driver program to test above function``arr ``=` `[``1``, ``2``, ``3``]``m ``=` `len``(arr)``n ``=` `4``x ``=` `count(arr, m, n)``print` `(x)` `# This code is contributed by Afzal Ansari`

## C#

 `// Dynamic Programming C# implementation``// of Coin Change problem``using` `System;` `class` `GFG``{``static` `int` `count(``int` `[]S, ``int` `m, ``int` `n)``{``    ``// table[i] will be storing the``    ``// number of solutions for value i.``    ``// We need n+1 rows as the table``    ``// is constructed in bottom up manner``    ``// using the base case (n = 0)``    ``int` `[] table = ``new` `int``[n + 1];``    ` `    ``// Base case (If given value is 0)``    ``table = 1;` `    ``// Pick all coins one by one and``    ``// update the table[] values after``    ``// the index greater than or equal``    ``// to the value of the picked coin``    ``for``(``int` `i = 0; i < m; i++)``        ``for``(``int` `j = S[i]; j <= n; j++)``            ``table[j] += table[j - S[i]];` `    ``return` `table[n];``}` `// Driver Code``public` `static` `void` `Main()``{``    ``int` `[]arr = {1, 2, 3};``    ``int` `m = arr.Length;``    ``int` `n = 4;``    ``Console.Write(count(arr, m, n));``}``}` `// This code is contributed by Raj`

## PHP

 ``

## Javascript

 ``

Output:

`4`

Thanks to Rohan Laishram for suggesting this space optimized version.
References:
http://www.algorithmist.com/index.php/Coin_Change

Following is another Top Down DP Approach:

## C++

 `#include ``#include ``using` `namespace` `std;` `int` `coinsExchange(``int` `amt, ``int` `*deno, ``int` `n, ``int` `*dp) {``    ``// base case``    ``if` `(amt == 0) {``        ``return` `dp[amt] = 0;``    ``}``    ``if` `(dp[amt] != -1) {``        ``return` `dp[amt];``    ``}` `    ``// recursive case``    ``int` `ans = INT_MAX;``    ``for` `(``int` `i = 0 ; i < n ; i++) {``        ``if` `(amt >= deno[i]) {``            ``int` `sa = coinsExchange(amt - deno[i], deno, n, dp);``            ``if` `(sa != INT_MAX) {``                ``ans = min(sa + 1, ans);``            ``}``        ``}``    ``}` `    ``return` `dp[amt] = ans;``}` `int` `main() {` `    ``int` `deno = {1, 2, 5, 10};``    ``int` `amt = 50;``    ``int` `*dp = ``new` `int``[amt + 1];``    ``for` `(``int` `i = 0 ; i <= amt ; i++) {``        ``dp[i] = -1;``    ``}``    ``cout << coinsExchange(amt, deno, 4, dp) << endl;` `    ``return` `0;``}`

Time Complexity: O(M*N)

Space Complexity: O(N)

Contributed by: Mayukh Sinha

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