Two dice are thrown independently. Let $$A$$ be the event that the number appeared on the $$1^{st}$$ die is less than the number appeared on the $$2^{nd}$$ die, $$B$$ be the event that the number appeared on the $$1^{st}$$ die is even and that on the second die is odd, and $$C$$ be the event that the number appeared on the $$1^{st}$$ die is odd and that on the $$2^{nd}$$ is even. Then

$$A = \{(x, y) \mid x < y\}$$

$$B = \{(x, y) \mid x \in \{2, 4, 6\} \text{ and } y \in \{1, 3, 5\}\}$$

$$C = \{(x, y) \mid x \in \{1, 3, 5\} \text{ and } y \in \{2, 4, 6\}\}$$

From definition of sets:

$$A \cap C = \{(1,2), (1,4), (1,6), (3,4), (3,6), (5,6)\} \implies n(A \cap C) = 6$$

$$B \cap C = \emptyset \implies n(B \cap C) = 0$$

Using distributive law:

$$(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$$

$$n((A \cup B) \cap C) = n(A \cap C) + n(B \cap C) - n(A \cap B \cap C) = 6 + 0 - 0 = 6$$