<p>You are given an integer array <code>coins</code> representing coins of different denominations and an integer <code>amount</code> representing a total amount of money.</p>

<p>Return <em>the number of combinations that make up that amount</em>. If that amount of money cannot be made up by any combination of the coins, return <code>0</code>.</p>

<p>You may assume that you have an infinite number of each kind of coin.</p>

<p>The answer is <strong>guaranteed</strong> to fit into a signed <strong>32-bit</strong> integer.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> amount = 5, coins = [1,2,5]
<strong>Output:</strong> 4
<strong>Explanation:</strong> there are four ways to make up the amount:
5=5
5=2+2+1
5=2+1+1+1
5=1+1+1+1+1
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> amount = 3, coins = [2]
<strong>Output:</strong> 0
<strong>Explanation:</strong> the amount of 3 cannot be made up just with coins of 2.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> amount = 10, coins = [10]
<strong>Output:</strong> 1
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= coins.length &lt;= 300</code></li>
	<li><code>1 &lt;= coins[i] &lt;= 5000</code></li>
	<li>All the values of <code>coins</code> are <strong>unique</strong>.</li>
	<li><code>0 &lt;= amount &lt;= 5000</code></li>
</ul>

class Solution {
    // int solve(  vector<int>& coins , int amount , int sum , int index){
    //     if(index == coins.size()){
    //         if(sum == amount) return 1;
    //         else return 0;
    //     }
    //     int includeCoin = solve(coins, amount, sum + coins[index], index);
    //     int skipCoin = solve(coins, amount, sum, index + 1);

    //     return includeCoin + skipCoin;
    // }

    int solve(vector<int>& coins,int amount,int sum,int index){

        if (sum == amount) return 1;  
        if (sum > amount || index == coins.size()) return 0;   

        int ways = 0;
        for (int i = index; i < coins.size(); ++i) {
            ways += solve(coins, amount, sum + coins[i], i); 
            
        }
        return ways;
       
    }

public:
    int t[201][5001];
    // int unbounded(vector<int>& coins,int amount,int n){

    //     if(amount == 0) return 1;
        
    //     if (n ==0) return 0;
    //     if (n <= 0 || amount < 0) return 0;
    //     if(t[n][amount] !=-1) return t[n][amount];
    //     if(coins[n-1]<=amount){
    //         return t[n][amount] = unbounded(coins,amount-coins[n-1],n) + unbounded(coins,amount,n-1);
    //     }else{
    //         return t[n][amount] = unbounded(coins,amount,n-1);
    //     }
    // }

    void solveTab(vector<int>& coins,int amount,int n,vector<vector<int>> &dp){

        for(int i=0;i<=n;i++){
            dp[i][0] = 1;
        }
        
        for(int i=1;i<=n;i++){
            for(int j=1;j<=amount;j++){
                if(coins[i-1]<=j){
                    dp[i][j] = dp[i][j-coins[i-1]] + dp[i-1][j];
                }
                else{
                    dp[i][j] = dp[i-1][j];
                }
            }
        }

    }
    int change(int amount, vector<int>& coins) {
        // return solve(coins,amount,0,0);
        int n = coins.size();
        // memset(t,-1,sizeof(t));
        vector<vector<int>> dp(n+1,vector<int>(amount+1,0));
        solveTab(coins,amount,n,dp);
        return dp[n][amount];
    }
};
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