<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>

def f1(denoms, amt, idx):
    if amt < 0:
        return 0
    if amt == 0:
        return 1

    ans = 0
    for i in range(idx, len(denoms)):
        denom = denoms[i]
        if denom <= amt:
            ans += f1(denoms, amt-denom, i)
    
    return ans

def f2(denoms, amt, idx, DP):
    if amt < 0:
        return 0
    if amt == 0:
        return 1
    if DP[amt][idx] != -1:
        return DP[amt][idx]
    ans = 0
    for i in range(idx, len(denoms)):
        denom = denoms[i]
        if denom <= amt:
            ans += f2(denoms, amt-denom, i, DP)
    DP[amt][idx] = ans
    return ans

def f3(denoms, amount):

    DP = [[0 for _ in range(len(denoms))] for _ in range(amount+1)]

    for amt in range(0, amount+1):
        for idx in range(len(denoms)-1, -1, -1):
            if amt == 0:
                DP[amt][idx] = 1
            else:
                ans = 0
                for i in range(idx, len(denoms)):
                    denom = denoms[i]
                    if denom <= amt:
                        ans += DP[amt-denom][i]
                DP[amt][idx] = ans

    return DP[amount][0]

class Solution:
    def change(self, amount: int, coins: List[int]) -> int:
        # res = f1(coins, amount, 0) 

        # DP = [[-1 for _ in range(len(coins))] for _ in range(amount+1)]
        # res = f2(coins, amount, 0, DP)

        res = f3(coins, amount)

        return res
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