Evaluate the following expression:
$$3\frac{2}{3}+2\frac{3}{4}+1\frac{1}{2}=?$$

Separate the whole number parts and the fractional parts:
$$\text{Sum}=(3+2+1)+\left(\frac{2}{3}+\frac{3}{4}+\frac{1}{2}\right)$$

$$\text{Whole part}=6$$

Find a common denominator for the fractional parts. The LCM of $3$, $4$, and $2$ is $12$:

$$\frac{2}{3}=\frac{8}{12}$$

$$\frac{3}{4}=\frac{9}{12}$$

$$\frac{1}{2}=\frac{6}{12}$$

Add the fractional parts together:

$$\frac{8}{12}+\frac{9}{12}+\frac{6}{12}=\frac{23}{12}=1\frac{11}{12}$$

Combine the whole part and the fractional part:

$$\text{Total Sum}=6+1\frac{11}{12}=7\frac{11}{12}$$