https://file.aiquickdraw.com/mathbot/file/1727319959863j69arwmx.jpg

From the image, the problem provided is about a survey of 116 high school students regarding their usage of Instagram, Twitter, or Facebook. Several pieces of information about the number of students using different combinations of these social media platforms are given, and the task is to answer questions related to this data.

Let’s break it down using a Venn diagram approach:

### Given Information:
- Total students = 116
- Instagram (I) = 71
- Twitter (T) = 60
- Facebook (F) = 75
- Instagram and Twitter (I ∩ T) = 35
- Instagram and Facebook (I ∩ F) = 43
- Twitter and Facebook (T ∩ F) = 40
- All three platforms (I ∩ T ∩ F) = 21

#### We need to determine the following:

### Part a) How many students surveyed used **only Instagram**?
- Using set operations: 
  - Those who used **Instagram and Twitter** but not Facebook: 35 − 21 = 14 (as given).
  - Those who used **Instagram and Facebook** but not Twitter: 43 − 21 = 22.
  - Those who used **only Instagram**: 71 − (14 + 22 + 21) = 14.
  - So, **14 students used only Instagram** (this matches the answer already provided).

### Part b) How many students surveyed used **Instagram and Twitter, but not Facebook**?
- **Instagram and Twitter** but **not Facebook**: 
  - This would be \( I \cap T \) but exclude \( I \cap T \cap F \). 
  - Already calculated as 35 − 21 = **14**.
  - So, **14 students used Instagram and Twitter but not Facebook** (this is also provided).

### Part c) How many students surveyed used **Instagram or Twitter**?
- For this, we need to compute the union of Instagram and Twitter:
  - \( |I \cup T| = |I| + |T| − |I \cap T| \).
  - So, \( |I \cup T| = 71 + 60 − 35 = 96 \).
  - Therefore, **96 students used Instagram or Twitter**.

Would you like further clarification or details on any of the steps? Let me know! Here are some follow-up questions that might help you think about this more deeply:

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1. How many students surveyed used only Twitter?
2. How many students surveyed used only Facebook?
3. How many students surveyed used only one platform (either Instagram, Twitter, or Facebook)?
4. How many students surveyed used **Instagram, but not Twitter or Facebook**?
5. How many students surveyed used **at least two platforms**?

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### Tip:
Always visualize sets with a Venn diagram when dealing with survey or intersection data. It helps to clearly track overlapping and unique groups.

A survey of 116 students determined their usage of Instagram, Twitter, and Facebook. The given data includes the number of students using each platform and various combinations of them. The questions involve calculating the number of students who used only Instagram, those who used Instagram and Twitter but not Facebook, and those who used either Instagram or Twitter.

Analyze a survey of 116 students on their usage of Instagram, Twitter, and Facebook. Learn how to calculate the number of students using only Instagram, those using Instagram and Twitter but not Facebook, and those using either Instagram or Twitter using set theory and the inclusion-exclusion principle.

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