Question 1177369
Let's solve this problem using the principle of inclusion-exclusion and Venn diagrams.

**1. Define Sets**

* **O:** Compositions using full orchestra (38)
* **P:** Compositions including piano (7)
* **V:** Compositions including solo violin (5)
* **C:** Compositions including chorus (7)
* **O_only:** Compositions for orchestra alone (21)

**2. Given Information**

* |O| = 38
* |P| = 7
* |V| = 5
* |C| = 7
* |P ∩ V| = 1
* |V ∩ C| = 0
* |O_only| = 21

**3. Find the Number of Compositions with at Least One of P, V, or C**

We know that:

* |O| = |O_only| + |P ∪ V ∪ C|

Therefore:

* |P ∪ V ∪ C| = |O| - |O_only| = 38 - 21 = 17

**4. Use the Inclusion-Exclusion Principle**

We have the formula for the union of three sets:

* |P ∪ V ∪ C| = |P| + |V| + |C| - |P ∩ V| - |P ∩ C| - |V ∩ C| + |P ∩ V ∩ C|

Plug in the known values:

* 17 = 7 + 5 + 7 - 1 - |P ∩ C| - 0 + |P ∩ V ∩ C|
* 17 = 18 - |P ∩ C| + |P ∩ V ∩ C|

**5. Find |P ∩ C| (Compositions with Piano and Chorus)**

Rearrange the equation:

* |P ∩ C| - |P ∩ V ∩ C| = 18 - 17
* |P ∩ C| - |P ∩ V ∩ C| = 1

We also know that |P ∩ V| = 1. This means that:

* |P ∩ V ∩ C| <= 1

Since |V ∩ C| = 0, we know |P ∩ V ∩ C| = 0. Therefore:

* |P ∩ C| - 0 = 1
* |P ∩ C| = 1

**Conclusion**

Yes, Ludwig wrote one composition that included both piano and chorus.