<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 {
public:
    int solve(int idx,int amt, vector<int>&coins,vector<vector<int>>&dp)
    {
       if(idx==coins.size()-1)
       {
         return (amt%coins[idx]==0);
       }

        if(dp[idx][amt]!=-1){
            return dp[idx][amt];
        }

        int not_pick=solve(idx+1,amt,coins,dp);
        int pick=0;
        if(coins[idx]<=amt)
        {
            pick=solve(idx,amt-coins[idx],coins,dp);
        }

        return dp[idx][amt]=pick+not_pick;
    }
    int change(int amount, vector<int>& coins) {
        int n=coins.size();
        vector<vector<int>> dp(n+1,vector<int>(amount+1,-1));

        // int sum=0;
        // for(int i=0;i<n;i++)
        // {
        //     sum=sum+coins[i];
        //}
        // if(sum<amount)
        // {
        //     return 0;
        // }

        return solve(0,amount,coins,dp);
    }
};
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