For any two sets S and T, S$\Delta$T is defined as the set of all elements that belong to either S or T but not both, that is, $S\Delta T=(S\cup T)-(S\cap T).$ Let A, B and C be sets such that $A\cap B\cap C=\{\}$, and the number of elements in each of A$\Delta$B, B$\Delta$C and C$\Delta$A equals 100. Then the number of elements in A$\cup$B$\cup$C equals:

Question

Accepted Answer

To determine the number of elements in $ A \cup B \cup C $, we need to analyze the given conditions and apply set theory principles.

### Explanation:

1. **Understanding the Symmetric Difference:**
   - The symmetric difference $ S \Delta T $ of two sets $ S $ and $ T $ is defined as the set of elements that are in either $ S $ or $ T $ but not in both. Mathematically, it is expressed as:
     $$
     S \Delta T = (S \cup T) - (S \cap T)
     $$
   - Given that $ |A \Delta B| = 100 $, $ |B \Delta C| = 100 $, and $ |C \Delta A| = 100 $.

2. **Using the Condition $ A \cap B \cap C = \{\} $:**
   - This condition implies that no element is common to all three sets $ A $, $ B $, and $ C $.

3. **Applying the Principle of Inclusion-Exclusion:**
   - The formula for the union of three sets is:
     $$
     |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C|
     $$
   - Since $ A \cap B \cap C = \{\} $, the term $ |A \cap B \cap C| $ is zero.

4. **Relating Symmetric Differences to Set Sizes:**
   - From the symmetric differences, we have:
     $$
     |A \Delta B| = |A| + |B| - 2|A \cap B| = 100
     $$
     $$
     |B \Delta C| = |B| + |C| - 2|B \cap C| = 100
     $$
     $$
     |C \Delta A| = |C| + |A| - 2|C \cap A| = 100
     $$

5. **Solving the Equations:**
   - Adding the three symmetric difference equations:
     $$
     (|A| + |B| - 2|A \cap B|) + (|B| + |C| - 2|B \cap C|) + (|C| + |A| - 2|C \cap A|) = 300
     $$
   - Simplifying gives:
     $$
     2(|A| + |B| + |C|) - 2(|A \cap B| + |B \cap C| + |C \cap A|) = 300
     $$
   - Dividing by 2:
     $$
     |A| + |B| + |C| - (|A \cap B| + |B \cap C| + |C \cap A|) = 150
     $$

6. **Conclusion:**
   - From the inclusion-exclusion principle and the above equation, we find:
     $$
     |A \cup B \cup C| = 150
     $$

Thus, the number of elements in $ A \cup B \cup C $ is 150, which corresponds to **Option 1**.

### Why Other Options are Incorrect:

- **Option 2 (300):** This would imply that the sets are completely disjoint and each symmetric difference contributes fully to the union, which contradicts the given symmetric difference values.
- **Option 3 (230):** This value does not satisfy the derived equation from the symmetric differences and inclusion-exclusion principle.
- **Option 4 (210):** Similar to Option 3, this does not align with the calculated result from the given conditions.

Therefore, the correct answer is indeed **Option 1 (150)**, as it is consistent with the mathematical derivation based on the problem's conditions.