1. **State the problem:** Multiply the mixed numbers $2$, $2 \frac{1}{3}$, and $1 \frac{1}{2}$. 2. **Convert mixed numbers to improper fractions:** $2 \frac{1}{3} = \frac{7}{3}$ because $2 \times 3 + 1 = 7$. $1 \frac{1}{2} = \frac{3}{2}$ because $1 \times 2 + 1 = 3$. 3. **Rewrite the multiplication:** $2 \times \frac{7}{3} \times \frac{3}{2}$. 4. **Convert whole number to fraction:** $2 = \frac{2}{1}$. 5. **Multiply all fractions:** $$\frac{2}{1} \times \frac{7}{3} \times \frac{3}{2} = \frac{2 \times 7 \times 3}{1 \times 3 \times 2} = \frac{42}{6}.$$ 6. **Simplify the fraction:** $\frac{42}{6} = 7$. 7. **Final answer:** $$7$$