Bowl A. Each bowl can hold B balls. There are C balls in total. How many ways are there (the balls are exactly the same)?

The bowl is the same, the ball is the same, change your thinking.
For the table in column A and row B, each steel ball can be placed in a grid, for a total of C steel balls.
Move the steel ball to create various possibilities. In order to avoid duplication and omission, we move the steel balls according to the following rules.
Place steel balls from left to right. After the first column is full, place the second column; and so on.
When moving the steel balls, always ensure that the number of steel balls in any column on the left is no less than that in any column on the right.
Every time a steel ball is moved, a possible situation occurs, and the counter increases by 1.

$boxNum = 20;
$size = 10;
$total = 33;

//Initialization
$data = array();
for($i=1;$i<=$boxNum;$ i++)
{
 if($total>=$size)
 {
 $data[$i] = $size;
 $total -= $size;
 }
 else if($total>0)
 {
 $ data[$i] = $total;
 $total = 0;
 }
 else
 {
 $data[$i] = 0;
 }
}

for($i=1;$i<=$boxNum ;$i++) echo $data[$i]."t";
echo "rn";



$count = 1;
while(true)
{
 for($i=$boxNum;$i> =1;$i--)
 {
 //Find the last value
 if($data[$i]>0)
 {
 $last = $i;
 break;
 }
 }

 list( $prev,$next) = getPrevNext($data,$last);
 if($prev===false)
 {
 if($last<$boxNum)
 {
 if($data[1]>1 )
 {
 list($prev,$next) = getPrevNext($data,$last+1);//Enable new column
 }
 else
 {
 break;//End
 }
 }
 else
 {
 break;//End
 }
 }

 $num = floor(($data[$prev] - $data[$next])/2);
 $data[$prev] -= $num;
 $data [$next] += $num;
 $count += $num;

 for($i=1;$i<=$boxNum;$i++) echo $data[$i]."t";
 echo "num:".$num."rn";
}


echo 'There are'.$count.' possibilities'."rn";


function getPrevNext($data,$last)
{
 $ prev = $next = false;
 for($i=$last-1;$i>=1;$i--)
 {
 if($data[$i]-$data[$i+1]> ;1)//First-order rolling found
 {
 $prev = $i;
 $next = $i+1;
 break;
 }
 else if($data[$i]-$data[$last] >1)//Multiple-order rolling is found
 {
 $prev = $i;
 for($k=$i+1;$k<=$last;$k++)
 {
 if($data[$ i]-$data[$k]>1)//Find the shortest multi-step roll-off
 {
 $next = $k;
 break;
 }
 }
 break;
 }
 }

 return array($ prev,$next);
}