Producing residues of combinatorics on strings
Think about alternatives


  Let's says I have 2 strings ABCDEF and ACDEBF. Now before I call gen8string, I will first perform a \$\binom 6 4\$ deletion operation on each string. For instance, I will take ABCDEF delete any two characters in that string and I am left with substring of length 4. How many different such strings can I have (given all the letters are unique)? \$15=\binom 6 4\$. Now, I will do the same for both strings. Now I am left with 2 string arrays, each of size 15.


Why?  

You want to generate 90 strings.  Why generate 255 strings and then mask out 165 of them?  Why not just generate 90 strings directly?  


  ABEFCDEF 001122
  ABDFCDEF 001212
  ABDECDEF 001221
  ...
  ABCFBDEF 020112
  ...
  ABEFABCD 221100


The six numbers are which string has the letter.  So 001122 means AB in the first string, CD in the second string, and EF in both.  Iterate from 001122 to 221100 or vice versa.  Then you don't need gen8string at all, as all it does is eliminate invalid strings like 2222-- and 22210-.  

Of course, this would happen outside the code that you posted.  

Working with what we have here

You have 


public static String[] gen8String(String[] pattern1, String[] pattern2){
    String[] combinedSubset = new String[90]; //emty array for the subset of 90 strings
    String  combinedString = ""; //string holder for each combined string
    int index = 0; //used for combinedSubset array
    int present = 0; //used to check if all 6 characters are present

    for(int i = 0; i &lt; 15; i++){

        for(int j = 0; j &lt; 15; j++){
            combinedString = pattern1[i] + pattern2[j]; //combine both 4 letter strings into 8 char length string
            char[] parsedString = combinedString.toCharArray(); //parse into array

            //check if all 6 characters are present
            for(int k = 1; k &lt;= 6; k++)
            {
                if(new String(parsedString).contains(k+"")) {
                    present++;
                }
                else
                    break;
                //if all 6 are present, then add it to combined subset
                if(present == 6)
                    combinedSubset[index++] = combinedString;
            }
            present = 0;
        }
    }
    return combinedSubset;
}



Consider 

public static String[] gen8String(String[] patterns1, String[] patterns2) {
    String[] results = new String[90];
    int resultCount = 0;

    for (String pattern1 : patterns1) {    
        for (String pattern2 : patterns2) {
            String combined = pattern1 + pattern2;

            //check if all 6 characters are present
            for (int k = 1; combined.contains(k+""); k++) {
                //if all 6 are present, then add it to the results and end
                if (k &gt;= 6) {
                    results[resultCount++] = combined;
                    break;
                }
            }
        }
    }

    return results;
}


This changes the names of the parameters.  They are collections of patterns, not the patterns themselves.  

I changed the names of combinedSubset and index to be clearer about how they relate to the results.  This allowed me to drop the comments as obsolete.  

Switching from C-style for loops to range based for loops gets rid of the unnecessary i and j variables.  It also gets rid of the magic number 15.  

I renamed combinedString to combined and moved the declaration to where it was set.  Since a single value is never used across iterations of the outer two loops, the scope doesn't need to be any wider.  

We don't need present, as we already have k.  If we make it to the sixth iteration, we can update the results and stop the loop.  

If we gate the loop on the contains check, we can just check the iterations once inside the loop.  The original code checked twice on each iteration.  

Note that your example suggested inputs like ABCD and CDEF.  The patterns here are actually like 1234 and 3456.  

Generalizing

Your code only works for four element subsets of a six element set.  Consider rewriting so the results are a List.  Then you wouldn't have to give a result set size or maintain your own count of the results.  

The largest value for k (6 here) should probably be a parameter.  Or pass in an array or list over which we could iterate.  

I already modified the for loops to be based on the size of the input parameters, so that's already generalized.  

And of course the name would need to change.  

Precalculate

Because the current results are so narrow, you could just set them manually.  One ninety element array.