If n(ABC) = 40, n(C) B'A') = 6, n(B) A'C') = 10, and n(A B'C') = 5 (where n(X) = number of elements in set X) then number of elements which belongs to at least two of the sets is 

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.

If n(ABC) = 40, n(C B'A') = 6, n(B A'C') = 10, and n(A B'C') = 5 (where n(X) = number of elements in set X) then number of elements which belongs to at least two of the sets is

Learn how to solve a set theory problem involving three sets, A, B, and C. The solution includes using the Inclusion-Exclusion Principle and calculating the number of elements belonging to at least two sets. Suitable for students in Grades 10-12.

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