I tried using a Venn diagram to understand this property but when I do so the middle portion $(A \cap B \cap C)$ repeats four times. So can Venn diagrams not be used for $n()$ properties? So is the above rule not the same as
$$(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)?$$
And if both are same and valid, can someone explain the Venn diagram?