How to implement such an algorithm?

Question

Say there are x number of boxes, each box contains an "inventory" of the letters A-Z, with each letter inventory being 1 or more.

Now say I need the following:

• 6 As
• 2 Bs
• 1 C

How do I get a list of all the possible combination/permutation of boxes that can provide me with the letters I need?

The algorithm needs to also produce combination of boxes to meet my requirements. For example: say Box-1 only has 4 As and Box-2 has 1 A and Box-3 has 1 A, I need the result of the algorithm to specify that the 6 As can be fulfilled across the 3 boxes.

What is the basic logic to the solution of such a problem. Are there any particular algorithms I need to be looking into for this?

EDIT 1:

Per dcp's suggestion, here's is my attempt that the PHP implementation:

$boxes = array(
    // box 1
    array(
        'A' => 6,
        'B' => 4,
        'C' => 10
    ),
    // box 2
    array(
        'A' => 8,
        'B' => 4,
        'C' => 2
    ),
    // box 3
    array(
        'A' => 1,
        'B' => 1,
        'C' => 0
    )
);

$n = count($boxes);

for ($mask = 0; $mask <= (1 << $n) - 1; $mask++)
{
    $tots = array();
    for ($i = 0; $i < $n; $i++)
    {
        if (((1 << $i) & $mask) != 0)
        {
            // this is a selected box for this mask, add the A's, B's etc. for this box to the total
            $tots[0] += $boxes[$i]['A'];
            $tots[1] += $boxes[$i]['B'];
            $tots[2] += $boxes[$i]['C'];
        }
        // check the tots array to see if it meets the letter requirements. If it does,
        // then this is a valid combination of boxes.
    }
}

Answer 1

If the number of boxes is fairly small, say 25 or less, then you could use a bitmask to brute force over all the possible box combinations:

// assume n is number of boxes, and boxes is the array of boxes
for(int mask = 0; mask <= (1<<n)-1; ++mask) {
  int tots[26];
  for(int i = 0; i < n; ++i) {
    if ( ((1<<i)&mask) != 0 ) {
      // this is a selected box for this mask, add the A's, B's etc. for this box to the total
      tots[0] += number of A's in box i
      tots[1] += number of B's in box i
      .
      .
    }
    // check the tots array to see if it meets the letter requirements. If it does, 
    // then this is a valid combination of boxes.
  }
}