<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 fewest number of coins that you need to make up that amount</em>. If that amount of money cannot be made up by any combination of the coins, return <code>-1</code>.</p>

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

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

<pre>
<strong>Input:</strong> coins = [1,2,5], amount = 11
<strong>Output:</strong> 3
<strong>Explanation:</strong> 11 = 5 + 5 + 1
</pre>

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

<pre>
<strong>Input:</strong> coins = [2], amount = 3
<strong>Output:</strong> -1
</pre>

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

<pre>
<strong>Input:</strong> coins = [1], amount = 0
<strong>Output:</strong> 0
</pre>

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

<ul>
	<li><code>1 &lt;= coins.length &lt;= 12</code></li>
	<li><code>1 &lt;= coins[i] &lt;= 2<sup>31</sup> - 1</code></li>
	<li><code>0 &lt;= amount &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution {
public:

    int findLowestCoins(vector<int>& coins, int index, int amount,
                        vector<vector<int>>& dp) {
        if (index >= coins.size() || amount <= 0) {
            if (amount == 0) {
                return 0;
            }
            return 1e9;
        }
        if (dp[index][amount] != -1)
            return dp[index][amount];

        int doNotTakeCoin = 1e9;
        int takeCoin = 1e9;
        if (coins[index] > amount) {
            doNotTakeCoin = 0 + findLowestCoins(coins, index + 1, amount, dp);
        } else {
            takeCoin =
                1 + findLowestCoins(coins, index, amount - coins[index], dp);
            doNotTakeCoin = 0 + findLowestCoins(coins, index + 1, amount, dp);
        }
        return dp[index][amount] = min(takeCoin, doNotTakeCoin);
    }
    int coinChange(vector<int>& coins, int amount) {
        int n = coins.size();
        vector<vector<int>> dp(n + 1, vector<int>(amount + 1, -1));
        int index = 0;
        int res = findLowestCoins(coins, index, amount, dp);
        if (res >= 1e9) {
            return -1;
        }
        return res;
    }
};
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