A survey of 111 high school students determined whether they used​ Instagram, Twitter, or Facebook. The provided information was determined.
72 used Instagram.
54 used Twitter.
75 used Facebook.
35 used Instagram and Twitter.
48 used Instagram and Facebook.
36 used Twitter and Facebook.
23 had all three features.
Complete parts ​a) through ​e). How many of the students surveyed used exactly two of the social​ medias?

To solve this problem, we can use the principle of inclusion-exclusion. Let's define the following:

- \( |I| = 72 \) (students who used Instagram)
- \( |T| = 54 \) (students who used Twitter)
- \( |F| = 75 \) (students who used Facebook)
- \( |I \cap T| = 35 \) (students who used both Instagram and Twitter)
- \( |I \cap F| = 48 \) (students who used both Instagram and Facebook)
- \( |T \cap F| = 36 \) (students who used both Twitter and Facebook)
- \( |I \cap T \cap F| = 23 \) (students who used all three social media)

We need to find the number of students who used exactly two of the social media platforms.

### 1. Students who used Instagram and Twitter but not Facebook:
This can be calculated as:
\[
|I \cap T \cap F'| = |I \cap T| - |I \cap T \cap F| = 35 - 23 = 12
\]

### 2. Students who used Instagram and Facebook but not Twitter:
This can be calculated as:
\[
|I \cap F \cap T'| = |I \cap F| - |I \cap T \cap F| = 48 - 23 = 25
\]

### 3. Students who used Twitter and Facebook but not Instagram:
This can be calculated as:
\[
|T \cap F \cap I'| = |T \cap F| - |I \cap T \cap F| = 36 - 23 = 13
\]

### Total number of students who used exactly two social media platforms:
\[
\text{Total} = |I \cap T \cap F'| + |I \cap F \cap T'| + |T \cap F \cap I'| = 12 + 25 + 13 = 50
\]

So, **50 students** used exactly two of the social media platforms.

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How many students used exactly one social media platform?
2. How many students used at least one social media platform?
3. How many students used exactly three social media platforms?
4. How many students used none of the social media platforms?
5. How can the principle of inclusion-exclusion be applied to other similar problems?

**Tip:** When solving problems with sets, Venn diagrams can be a helpful visual tool to organize and simplify the data.

Explore the analysis of a survey where 111 high school students used Instagram, Twitter, or Facebook. Discover how many students used exactly two social media platforms using the principle of inclusion-exclusion. Detailed step-by-step solution provided.

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