Conditional probability
Solution 1:
Yes you can. I see no fault. Because if you put $K = B \cap C$ you obtain the original result
Solution 2:
Yes and this is also known as multiple conditioning. In general, $$P(A_1 \cap \cdots \cap A_n) = P(A_1) \prod_{i=2}^{n} P(A_{i}|A_{1} \cap \cdots \cap A_{i-1})$$