This get the number of ways you can count an array of integers given the sum

how_many_ways.php

<?php

// This shows how many ways we can arrive at the amount using the given coins
function how_many_ways($m, $coins)
{
    // Use memoization to avoid repeated calculations
    $memo = [];
    $memo[0] = 1;
    $chain = [];

    for($i = 1; $i <= $m; $i++)
    {
        foreach($coins as $coin)
        {
            $diff = $i - $coin;

            if($diff < 0)
            {
                break;
            }

            if($diff >= 0 && isset($memo[$diff]))
            {
                $memo[$i] = ($memo[$i] ?? 0) + $memo[$diff];
                $chain[$coin][] = $diff;
            }
        }
    }

    // To get the coins used to make up the amount, we can use the following code
    $m_copy = $m;
    foreach($chain as $coin => $ways)
    {
        $chain[$coin] = [];
        $m = $m_copy;

        // Get the last coin used
        $prev_coin = $ways[count($ways) - 1];

        while($m > 0)
        {
            $m -= $coin;

            if($m < 0)
            {
                $chain[$coin][] = $prev_coin;
                break;
            }
            
            $chain[$coin][] = $coin;
        }
    }

    return [
        'amount' => $m_copy,
        'ways' => $memo[$m] ?? 0,
        'chain' => $chain,
    ];
}

print_r(
    how_many_ways(5, [1, 4, 5])
);

amount -> The target sum
ways -> The number of ways we can reach the target sum
chain -> simply shows a chain of how we can reach the target

For the example above, the output will be

Array
(
    [amount] => 5
    [ways] => 1
    [chain] => Array
        (
            [1] => Array
                (
                    [0] => 1
                    [1] => 1
                    [2] => 1
                    [3] => 1
                    [4] => 1
                )

            [4] => Array
                (
                    [0] => 4
                    [1] => 1
                )

            [5] => Array
                (
                    [0] => 5
                )

        )

)

We can read the chain as

1. We can have 1 + 1 + 1 + 1 + 1
2. Or 4 + 1
3. Or simply 5