or the compound:

carbon in the middle with 3 bonds
-double bond to oxygen
-bond to Mn(CO)5
-bond to CH3 group

is the oxidation state of the metal +1?
d count 6?
total electron count 20?
coordination number 7?

To determine the oxidation state of the metal in the compound, we need to consider the charges and electron distribution. Let's break down the compound and analyze it step by step:

1. Carbon in the middle with 3 bonds (-double bond to oxygen, -bond to Mn(CO)5, -bond to CH3 group): Carbon is generally assumed to have an oxidation state of -4 when it forms bonds with more electronegative elements like oxygen. So, with three bonds present, the carbon's oxidation state will be -4.

2. Double bond to oxygen: Oxygen is more electronegative than carbon, so it is assumed to have a -2 oxidation state. A double bond to oxygen implies a charge of -4 on the oxygen atom.

3. Bond to Mn(CO)5: In manganese pentacarbonyl, Mn(CO)5, the oxidation state of the manganese (Mn) is known to be zero (0). Each carbonyl (CO) ligand contributes a -2 charge, and since there are five CO ligands, the total charge from the CO ligands is -10. To balance that, the oxidation state of the metal must be +10 (the sum of all ligands' charges).

4. Bond to CH3 group: The methyl (CH3) group is neutral and does not affect the oxidation state of the metal.

Based on this analysis, the oxidation state of the metal (Mn) is indeed +1. The compound you described would have a Mn atom with an oxidation state of +1.

As for the other counts:
- d-count: The d-count refers to the number of d-electrons in the metal's valence shell. In this case, since the oxidation state of the metal is +1, we need to check the electron configuration of Mn+1. Mn is in the 4th period of the periodic table and has the electron configuration [Ar] 3d5 4s2. When Mn loses one electron (+1 oxidation state), it loses the 4s electron first, resulting in an electron configuration of [Ar] 3d5. Therefore, the d-count is 5.
- Total electron count: To calculate the total electron count, we sum up the valence electrons of each atom in the compound. The carbon in the middle has 4 valence electrons, the oxygen from the double bond has 6 valence electrons, manganese has 7 valence electrons (since its oxidation state is +1), and the methyl group contributes 4 valence electrons. Adding those up, we get a total of 21 valence electrons.
- Coordination number: The coordination number refers to the number of bonds or ligands attached to the central atom. In this case, the coordination number is 6 because there are a total of 6 bonds (3 bonds to carbon, 1 double bond to oxygen, 1 bond to Mn(CO)5, and 1 bond to the CH3 group) attached to the central carbon atom.

So, to summarize:
- The oxidation state of the metal (Mn) is +1.
- The d-count is 5.
- The total electron count is 21.
- The coordination number is 6.