1. **Stating the problem:** We have two sets $A$ and $B$. - $|A| = 12$ (number of elements in $A$) - $|B| = 12$ (number of elements in $B$) - $|A \cup B| = 21$ (number of elements in the union of $A$ and $B$) We need to find $|A \cap B|$ (number of elements in the intersection of $A$ and $B$). 2. **Formula used:** The formula relating the sizes of two sets and their union and intersection is: $$|A \cup B| = |A| + |B| - |A \cap B|$$ 3. **Substitute the known values:** $$21 = 12 + 12 - |A \cap B|$$ 4. **Simplify the equation:** $$21 = 24 - |A \cap B|$$ 5. **Solve for $|A \cap B|$:** $$|A \cap B| = 24 - 21 = 3$$ 6. **Interpretation:** The number of elements common to both sets $A$ and $B$ is 3. 7. **Check the options:** - a. 6 - b. 8 - c. 4 - d. None of these Since 3 is not listed, the correct answer is **d. None of these**.