Find ordered combinations of search string letters in a haystack string Hacker Rank challenge

Question

I attempted to pass a Hacker Rank test, but I failed. The test asked you to count the number of $newWords there are in a $oldWord. The letters must be in the correct order, even though they can have other characters between them. 

Example:

$newWord = 'abc';
$oldWord = 'abcababc';

How many occurency of abc exist here?

The answer is 7:

abcababc
^^^
abcababc
^^     ^
abcababc
^   ^  ^
abcababc
^     ^^
abcababc
   ^^  ^
abcababc
   ^  ^^
abcababc
     ^^^

I have tried many things like splitting into an array, substr(), for() loops, and array_count_values(), but I have not found the solution.

I don't understand the logic. For example, how can I find abcababc? What I tried:

• Split word into equal parts (from count $newWord) then count with array values but I only find two ways.
• Remove one letter at a time and count $newWord.

This's another example:

$newWord = 'ccc';
$oldWord = 'cccc';

How many occurrences of ccc exist here?

The answer is 4:

cccc
^^^
cccc
^^ ^
cccc
^ ^^
cccc
 ^^^

Can someone please help me with this?