Question 1177369: Ludwig published 38 compositions that used the full orchestra. 7 of these included piano, 5 included solo violin and 7 included a chorus. Of these,just 1 included both piano and solo violin and none included both chorus and a solo violin. There were 21 compositions for orchestra alone (no solo instrument or chorus). Did he write any that included both piano and chorus? If so how many? Explain and show work.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
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