Total number of possible stereoisomers of the following co-ordinate compound Mabcdef where - a, b, c, d, e and f are mono dentate ligands.

To determine the total number of possible stereoisomers for the coordinate compound $Mabcdef$, where $a, b, c, d, e,$ and $f$ are monodentate ligands, we can follow these analytical steps:

1. ### Break Down the Problem

1. Identify the coordination number of the metal center $M$.
2. Determine the arrangement of the six monodentate ligands around the metal center.
3. Analyze geometric and optical isomerism possibilities.

2. ### Relevant Concepts

1. The coordination number determines how many ligands can surround the metal, which in this case is 6 (since there are six ligands).
2. The geometry of coordination compounds with a coordination number of 6 is typically octahedral.
3. The formula for calculating the number of stereoisomers for octahedral coordination compounds can be complex but typically involves considering facial (fac) and meridional (mer) arrangements, as well as any possible chiral arrangements.

3. ### Analysis and Detail

1. Chiral Arrangement: Since all ligands are unique, they can produce optical isomers.

2. Different Configurations:

   • Facial (fac): 3 ligands occupy one face of the octahedron, while the other 3 occupy the opposite face.
   • Meridional (mer): Ligands are arranged such that 2 ligands are on one side, and 2 on the opposite side, with a single ligand connecting these arrangements.

3. For six different ligands in an octahedral compound, we can establish stereoisomers through the following formula:

   • For a coordination compound with $n$ different monodentate ligands, the number of stereoisomers can be calculated as: $\text{Stereoisomers} = \frac{n!}{r!}$ where $n$ is the number of ligands and $r$ is the number of sets of equivalent ligands.

   For $Mabcdef$ (where $n = 6$):

   • Each of the 6 ligands is different, so $r = 1$ leading to $\text{Stereoisomers} = 6! = 720.$

4. ### Verify and Summarize

To sum up, with six unique monodentate ligands arranged around an octahedral metal center, we can conclude that $Mabcdef$ can yield:

Final Answer

720 possible stereoisomers.