Question 1177748: a class of 26 boys were each require to have certain text books in english french and mathematics. 19 boys had the english books, 23 had the french books and 15 had the mathematics. 16 boys had both the english and french books, 14 had the french and mathematics and 13 had both mathematics and the english books. illustrate on a venn diagram.
how many boys had all the three books.
two books only.
english and french but not mathematics.
only french books.
only one books.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step using the principle of inclusion-exclusion and Venn diagrams.
**1. Define Sets**
* E = boys with English books (19)
* F = boys with French books (23)
* M = boys with Mathematics books (15)
**2. Given Information**
* |E| = 19
* |F| = 23
* |M| = 15
* |E ∩ F| = 16
* |F ∩ M| = 14
* |M ∩ E| = 13
* Total boys = 26
**3. Find the Number of Boys with All Three Books (|E ∩ F ∩ M|)**
We can use the formula for the union of three sets:
* |E ∪ F ∪ M| = |E| + |F| + |M| - |E ∩ F| - |F ∩ M| - |M ∩ E| + |E ∩ F ∩ M|
We know that |E ∪ F ∪ M| is the total number of boys, which is 26.
* 26 = 19 + 23 + 15 - 16 - 14 - 13 + |E ∩ F ∩ M|
* 26 = 57 - 43 + |E ∩ F ∩ M|
* 26 = 14 + |E ∩ F ∩ M|
* |E ∩ F ∩ M| = 26 - 14 = 12
Therefore, 12 boys had all three books.
**4. Find the Number of Boys with Exactly Two Books**
* Only English and French: |E ∩ F| - |E ∩ F ∩ M| = 16 - 12 = 4
* Only French and Mathematics: |F ∩ M| - |E ∩ F ∩ M| = 14 - 12 = 2
* Only Mathematics and English: |M ∩ E| - |E ∩ F ∩ M| = 13 - 12 = 1
Total boys with exactly two books: 4 + 2 + 1 = 7
**5. Find the Number of Boys with Only English and French but Not Mathematics**
* This is the same as the number of boys with only English and French, which we calculated in step 4: 4.
**6. Find the Number of Boys with Only French Books**
* |F| - (boys with E and F) - (boys with F and M) + (boys with all three)
* |F| - (|E ∩ F| - |E ∩ F ∩ M|) - (|F ∩ M| - |E ∩ F ∩ M|) - |E ∩ F ∩ M| = 23 - 4 - 2 - 12 = 5
**7. Find the Number of Boys with Only One Book**
* Only English: |E| - (boys with E and F) - (boys with E and M) + (boys with all three) = 19 - 4 - 1 - 12 = 2
* Only French: 5
* Only Mathematics: |M| - (boys with M and E) - (boys with M and F) + (boys with all three) = 15 - 1 - 2 - 12 = 0
Total boys with only one book: 2 + 5 + 0 = 7
**Venn Diagram Illustration**
* Place 12 in the center (E ∩ F ∩ M).
* Place 4 in E ∩ F only.
* Place 2 in F ∩ M only.
* Place 1 in M ∩ E only.
* Place 2 in E only.
* Place 5 in F only.
* Place 0 in M only.
**Answers**
* Boys with all three books: 12
* Boys with two books only: 7
* Boys with English and French but not Mathematics: 4
* Boys with only French books: 5
* Boys with only one book: 7
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