Not a clue about the biology and Id not really think of it nCr ((n+1)C2 more precisely). Its simply as Icosagon demonstrated
AA, AB, AC, AD, AE
AB, BB, BC, BD, BE
CA, CB, CC, CD, CE
DA, DB, DC, DD, DE
EA, EB, EC, ED, EE,
The bolds are repeats which you dont want to count but the diagnonls only occur once. So think of it as a rectangle n*(n+1) where the diagonals are repeated, then simply divide by 2, so overall n(n+1)/2
AA, AA, AB, AC, AD, AE
AB, BB, BB, BC, BD, BE
CA, CB, CC, CC, CD, CE
DA, DB, DC, DD, DD, DE
EA, EB, EC, ED, EE, EE,