We are given the following information about three sets $A$, $B$, and $C$:

• $n(A \cup B \cup C) = 40$, which is the total number of elements in the union of the three sets.
• $n(C \cap B' \cap A') = 6$, which is the number of elements in $C$ but not in $B$ or $A$.
• $n(B \cap A' \cap C') = 10$, which is the number of elements in $B$ but not in $A$ or $C$.
• $n(A \cap B' \cap C') = 5$, which is the number of elements in $A$ but not in $B$ or $C$.

We need to find the number of elements that belong to at least two of the sets $A$, $B$, and $C$.

Solution Outline:

1. First, we calculate the number of elements that belong to exactly one set.
   • These are the elements in $A$ only, $B$ only, and $C$ only.
2. From the total number of elements $n(A \cup B \cup C)$, we subtract the number of elements that belong to exactly one set to find those that belong to at least two sets.

Step 1: Calculate number of elements in exactly one set

The number of elements in exactly one of the sets is: $n(A \cap B' \cap C') + n(B \cap A' \cap C') + n(C \cap A' \cap B') = 5 + 10 + 6 = 21$

Step 2: Calculate elements in at least two sets

The number of elements in at least two of the sets is: $n(A \cup B \cup C) - \text{(number of elements in exactly one set)}$ $= 40 - 21 = 19$

Final Answer:

The number of elements that belong to at least two of the sets is 19.

Would you like a further explanation of any steps or concepts?

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Here are 5 related questions you might find useful:

1. What is the formula for the union of three sets?
2. How can we find the number of elements in exactly two sets?
3. Can you calculate the number of elements that belong to all three sets using a similar method?
4. How do Venn diagrams help in visualizing problems like this?
5. How would the answer change if the total number of elements increased?

Tip: When dealing with set problems, breaking them into regions using complements helps manage complex intersections efficiently.