Question 1177748
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