Question 1172902
Absolutely, let's solve this problem step-by-step using a Venn diagram.

**1. Venn Diagram**

Let's use the following notation:

* F = Football
* B = Basketball
* C = Cricket

We are given:

* Total students = 50
* n(F) = 24
* n(B) = 21
* n(F ∩ B) = 6
* Students who like basketball only = 3
* n(F ∩ B ∩ C) = 5
* Students who like none of the games = 14

Here's how we'll build the Venn diagram:

1.  **Start with the intersection of all three:**
    * n(F ∩ B ∩ C) = 5. Place 5 in the center of the Venn diagram where all three circles overlap.

2.  **Football and Basketball:**
    * n(F ∩ B) = 6. We know 5 like all three, so 6 - 5 = 1 student likes only football and basketball. Place 1 in the F ∩ B region.

3.  **Basketball Only:**
    * 3 students like only basketball, place a 3 in the B only section of the venn diagram.

4.  **Basketball circle:**
    * We know n(B)=21. We have 5+1+3=9 of those students accounted for. 21-9=12. We do not yet know how many of those 12 like cricket, so we will fill in the values that we can.

5.  **Students who like none:**
    * 14 students like none of the games. Place 14 outside the circles.

6.  **Total within the circles:**
    * 50 total students - 14 who like none = 36 students who like at least one sport.

7.  **Football circle:**
    * n(F) = 24. We know 5+1=6 of those. 24-6=18. 18 students like football and cricket, or football only. We cannot yet determine the individual values.

8.  **Cricket values:**
    * We can find the total amount of students that like cricket. 36 students total like at least one sport. 36 - 3 - 1 - 5 - 18 = 9. 9 students like cricket only or cricket and football. We cannot yet determine the individual values.

9.  **Completing the diagram:**
    * We know that the remaining students that like basketball must total 12. Let x= the students that like basketball and cricket only. 12 = x+5. x=7. 7 students like Basketball and Cricket only.
    * We know the students that like cricket total 9+7+5 = 21.
    * We know that the football only students equal 24-1-5-students that like football and cricket only. 24-6-cricket and football only = football only. 21-5-7=9. 9 students like cricket only. 24-1-5 = 18. 21-9-5-7=0. 18-0=18. 18 students like football only.
    * The Venn diagram is complete.

**2. Answers**

**A. Football and Cricket:**

* To find the number of students who like football and cricket, we add the students who like only football and cricket (0) and those who like all three (5): 0+5=5.
* 5 students like football and cricket.

**B. Exactly One Game:**

* Add the number of students who like only football (18), only basketball (3), and only cricket (9): 18 + 3 + 9 = 30.
* 30 students like exactly one game.

**C. Exactly Two Games:**

* Add the number of students who like football and basketball only (1), basketball and cricket only (7), and football and cricket only (0): 1 + 7 + 0 = 8.
* 8 students like exactly two games.