Count the number of subsets of a set whose intersect with another set

Question

I have a question in the lecture that was answered by the lecturer. But then again, it was just briefly explained and i still cannot digest the idea

The question is count the subsets of the set X = {A,B,C,D,E,F,G} whose intersection with {A,B,C,D} has size 2 and the answer is 4C2 x 8 = 48. I was thinking about it but do not really understand why the first term relates to the second and it makes me so confused and wonder that “what am i counting now”

Also, do you have any general method to solve this kind of intersection of subsets ?

Thanks so much for your help in advance

Answer 1

You can construct each possible set $S$ that satisfies the requirements by first choosing $Y = S \cap \{A, B, C, D\}$ and then choosing $Z = S \cap \{E, F, G\}$. Distinct choices of $Y$ or $Z$ lead to distinct $S$. There are $_4C_2 = 6$ choices for $Y$ and $2^3 = 8$ choices for $Z$ leading to $6 \times 8 = 48$ possibilities for $S$.