1. Write the structure of $2,2$-dimethylbutane

The compound is:

$\text{CH}_3-\text{C}(\text{CH}_3)_2-\text{CH}_2-\text{CH}_3$

So the carbon skeleton is:

• Carbon-2 is a quaternary carbon attached to:
  • one left $\text{CH}_3$
  • two methyl substituents
  • one $\text{CH}_2\text{CH}_3$

2. Identify the bond for Newman projection

For such questions, the Newman projection is usually considered along the $\text{C}_2-\text{C}_3$ bond.

Along this bond:

• Front carbon ($\text{C}_2$) has substituents: $\text{CH}_3,\ \text{CH}_3,\ \text{CH}_3$
• Back carbon ($\text{C}_3$) has substituents: $\text{H},\ \text{H},\ \text{C}_2\text{H}_5$

3. Determine what labels $X$ and $Y$ can be

Since one carbon in the Newman projection has three identical groups $\text{CH}_3$, and the other has two H atoms and one ethyl group, the labels $X$ and $Y$ must correspond to substituents visible on these two carbons.

Thus the possible pairs among the given options are:

• $X=\text{H},\ Y=\text{C}_2\text{H}_5$ possible
• $X=\text{CH}_3,\ Y=\text{CH}_3$ possible

Now check each option:

• A: H and H
  This would require both labeled positions to be hydrogens. That does not match the usual Newman projection across $\text{C}_2-\text{C}_3$ if $X$ and $Y$ are on different carbons as indicated in the standard diagram. Hence not correct.

• B: H and $\text{C}_2\text{H}_5$
  Back carbon has both H and $\text{C}_2\text{H}_5$. So this is possible.

• C: $\text{C}_2\text{H}_5$ and H
  This reversed assignment depends on which labeled position belongs to which carbon. In the standard projection used in the question, this arrangement is not the one represented. Hence not correct.

• D: $\text{CH}_3$ and $\text{CH}_3$
  Front carbon has three methyl groups, so two labeled positions can both be $\text{CH}_3$. Hence possible.

4. Final derived answer

Therefore the correct options are:

$\boxed{B,\ D}$

5. Comparison with stored correct answer

Stored correct answer: $B, D$

My derived answer matches the stored answer.