Analytical formula for determining number of sequences which have some overlap with other sequences

Hello

I can create a number of unique sequences with say combnk(1:6,4) which gives

     3     4     5     6
     2     4     5     6
     2     3     5     6
     2     3     4     6
     2     3     4     5
     1     4     5     6
     1     3     5     6
     1     3     4     6
     1     3     4     5
     1     2     5     6
     1     2     4     6
     1     2     4     5
     1     2     3     6
     1     2     3     5
     1     2     3     4

but the number sequences that have at most 2 numbers in common is (ie a max of two numbers in common) is

3 4 5 6

1 2 5 6

1 2 3 4

I need an analytical formula for determining the number of such sequences. So we know 6C4 = combnk(1:6,4)gives the total number of unique sequences of four numbers (that differ from all other sequences by at least one number). What would be the formula for determining the number sequences that have atmost "x" number of numbers in common with all other sequences

Thanks in advance

[Cedric]

What about

 1 2 3 5
 1 2 4 6
 3 4 5 6

or

 2 3 4 5
 1 2 3 6
 1 4 5 6

or

 ...