Solve Four-Digit Addition: 4652 + 2161 in Vertical Format

$\begin{aligned} &4652 \\ +& \\ &2161 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

To solve this problem, we will utilize the vertical addition method as follows:

First, write the numbers one below the other, aligning them by their respective place values:

$\begin{aligned} &4652 \\ +& \\ &2161 \\ &\underline{\phantom{776}} \\ \end{aligned}$

Now, let's add the numbers column by column starting from the rightmost side (units place):
1. **Units Column**: $2 + 1 = 3$. Write 3 in the units place.
2. **Tens Column**: $5 + 6 = 11$. Write 1 in the tens place and carry over 1 to the hundreds place.
3. **Hundreds Column**: $6 + 1 + 1 = 8$. Write 8 in the hundreds place.
4. **Thousands Column**: $4 + 2 = 6$. Write 6 in the thousands place.

The sum is arranged as follows:

$\begin{aligned} &4652 \\ +& \\ &2161 \\ &\underline{6813} \\ \end{aligned}$

Therefore, the result of adding $4652$ and $2161$ is $\textbf{6813}$.

Checking against the choices provided, the correct answer is Choice 4: $6813$.