Step-by-Step Logical Deduction

To answer the question, we must first determine exactly which activities each of the five students subscribed to by systematically applying all the given rules and information.

1. Initial Deductions and Assignments

We can use a table to track the subscriptions. Let's fill it out based on the clues.

• Leena: She subscribes to Gymnastics but not Karate.
  • Rule "Gymnastics must also subscribe to Dance" means Leena takes Gymnastics and Dance.
  • Rule "Karate and Handwork must be subscribed together" means since she doesn't take Karate, she also doesn't take Handwork.
  • Rule "Music and Dance cannot be subscribed together" means since she takes Dance, she cannot take Music.
  • Leena's Activities: {Dance, Gymnastics}. (Total: 2 activities)
• Megan: She subscribes to 4 of the 5 activities.
  • Since Music and Dance cannot be taken together, she cannot take both. The 4 activities she takes cannot be {K,H,M,D}.
  • The only combination of 4 activities that works with the rules is {Karate, Handwork, Dance, Gymnastics}.
• Pixie: She subscribes to only one activity, and no one else subscribes to it.
  • Looking at the activities of Leena and Megan, the only activity not taken by either of them so far is Music.
  • Therefore, Pixie must subscribe to Music, and she is the only one who does.
  • Pixie's Activity: {Music}.
• Omar & Neha: Omar takes 3 activities, Neha takes 1.
  • They cannot take Music (unique to Pixie). Their activities must be chosen from {Karate, Handwork, Dance, Gymnastics}.
  • Omar takes 3. He must take the (Karate, Handwork) block. To take a 3rd, he can take Dance or Gymnastics. If he takes Gymnastics, he must also take Dance (making 4 activities), which is too many. So he must take Dance.
  • Omar's Activities: {Karate, Handwork, Dance}.
  • Neha takes 1 activity from the remaining pool. She cannot take the (Karate, Handwork) block. If she takes Gymnastics, she must also take Dance (2 activities). So, she must take only Dance.
  • Neha's Activity: {Dance}.

2. Final Subscription Count per Activity

Now we can count how many students are in each activity:

• Karate: Megan, Omar → 2 students
• Handwork: Megan, Omar → 2 students
• Music: Pixie → 1 student
• Dance: Leena, Megan, Neha, Omar → 4 students
• Gymnastics: Leena, Megan → 2 students

Conclusion

The question asks how many activities are subscribed to by exactly two people. From our count, the activities with exactly two subscribers are Karate, Handwork, and Gymnastics.

Therefore, there are 3 such activities.