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How to find number of isomers of a octahedral complex containing all different ligands for example ML2,L2,L3,L4,L5,L6 type ?

Let us consider a octahedral complexes [ML1L2L3L4L5L5]  where M is the central metal atom while L2, L2, L3 .... are six different ligands we  shall represent the ligands by smallcase letters a,b,c etc. and the metal by M. We can write 12"cis" isomers in which ligands are separated by at angle 90° and 3 "trans" isomers in which ligands are separated by 180°.

The geometrical isomers of Mabcdef areas follows.

M(L1)(L2)(L3) (L4)(L5)(L6):

"Cis" isomers: separated by 90°

(1,2)(1,3)(1,4)(1,5)

(2,3)(3,4)(4,5)(5,2)

(2,6)(3,6)(4,6)(5,6)

"Trans " isomers: separated by 180°

(1,6)(2,4)(3,5)



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