How many sigma $(\sigma)$ and pi $(\pi)$ bonds are in a molecule of ethyne (acetylene), $\ce{HCCH}?$

$ \textbf{(A) } 1 \sigma \text{ and } 1 \pi \qquad\textbf{(B) }2 \sigma \text{ and } 1 \pi \qquad\textbf{(C) }2 \sigma \text{ and } 3\pi \qquad\textbf{(D) }3 \sigma \text{ and } 2 \pi\qquad$
1. **Identify the structure of ethyne (acetylene), $\ce{HCCH}$:**
   - Ethyne is a linear molecule with the structure $\ce{H-C≡C-H}$.
   - Each carbon atom forms a triple bond with the other carbon atom and a single bond with a hydrogen atom.

2. **Determine the types of bonds in ethyne:**
   - A single bond (sigma bond, $\sigma$) exists between each hydrogen and carbon atom.
   - A triple bond between the two carbon atoms consists of one sigma bond and two pi bonds ($\pi$).

3. **Count the sigma bonds:**
   - There are two sigma bonds between the hydrogen atoms and the carbon atoms.
   - There is one sigma bond in the triple bond between the two carbon atoms.
   - Total sigma bonds: $2 + 1 = 3$.

4. **Count the pi bonds:**
   - The triple bond between the two carbon atoms contains two pi bonds.
   - Total pi bonds: $2$.

Therefore, the molecule of ethyne ($\ce{HCCH}$) contains 3 sigma bonds and 2 pi bonds.

The final answer is $\boxed{3 \sigma \text{ and } 2 \pi}$.