The total number of isomer including stereosisomers for 1,2-dimethyl cyclobutane is:

Options

A. (a) 2, one cis and one trans (both optically inactive)
B. (b) 3, one cis and two opticaly active trans froms
C. (c) 3, one trans and two optically active cis froms
D. (d) 4, two cis (optically active) and two trans (optically active)

Answer

To determine the total number of isomers, including stereoisomers, for 1,2-dimethylcyclobutane, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Structure**:
- Start by drawing the structure of cyclobutane, which is a four-membered ring.
- Then, add two methyl (CH3) groups at the 1 and 2 positions of the cyclobutane.

2. **Draw the Isomers**:
- There are two possible configurations for the methyl groups: they can be on the same side (cis) or on opposite sides (trans) of the cyclobutane ring.

3. **Cis Isomer**:
- The cis isomer has both methyl groups on the same side of the ring.
- This configuration has a plane of symmetry and is optically inactive.

4. **Trans Isomers**:
- The trans isomer has the methyl groups on opposite sides of the ring.
- There are two distinct trans configurations: one where the methyl groups are oriented in a way that they are non-superimposable mirror images of each other, resulting in two enantiomers.
- These two trans configurations do not have a plane of symmetry and are optically active.

5. **Count the Isomers**:
- From the above analysis, we have:
- 1 cis isomer (optically inactive)
- 2 trans isomers (optically active, which are enantiomers)
- Therefore, the total number of isomers is 3 (1 cis + 2 trans).

### Final Answer:
The total number of isomers, including stereoisomers, for 1,2-dimethylcyclobutane is **3**.