A survey of 185 UNLV students during the fall semester revealed that:
95 had Facebook accounts
116 had Instagram accounts
47 had Twitter accounts
29 had both Facebook & Twitter accounts
12 had both Instagram & Twitter accounts
74 had both Facebook & Instagram accounts
9 had all three social media accounts
33 did not have any of these social media accounts
Which diagram below is an accurate Venn diagram for the above information?

Question

UpStudy AI · Accepted Answer

We need to analyze the provided data to determine the number of students in each category of social media accounts. The information given includes the total number of students and the overlaps between different social media accounts.
Let:
- $$ F $$ = number of students with Facebook accounts = 95
- $$ I $$ = number of students with Instagram accounts = 116
- $$ T $$ = number of students with Twitter accounts = 47
- $$ F \cap T $$ = number of students with both Facebook and Twitter = 29
- $$ I \cap T $$ = number of students with both Instagram and Twitter = 12
- $$ F \cap I $$ = number of students with both Facebook and Instagram = 74
- $$ F \cap I \cap T $$ = number of students with all three accounts = 9
- Students with no accounts = 33
We will calculate the number of students who have only one or two accounts using the principle of inclusion-exclusion.

1. Students with only Facebook:
   $$
   F_{only} = F - (F \cap I) - (F \cap T) + (F \cap I \cap T)
   $$

2. Students with only Instagram:
   $$
   I_{only} = I - (F \cap I) - (I \cap T) + (F \cap I \cap T)
   $$

3. Students with only Twitter:
   $$
   T_{only} = T - (F \cap T) - (I \cap T) + (F \cap I \cap T)
   $$

4. Students with only Facebook and Instagram:
   $$
   F \cap I_{only} = (F \cap I) - (F \cap I \cap T)
   $$

5. Students with only Facebook and Twitter:
   $$
   F \cap T_{only} = (F \cap T) - (F \cap I \cap T)
   $$

6. Students with only Instagram and Twitter:
   $$
   I \cap T_{only} = (I \cap T) - (F \cap I \cap T)
   $$

Now, let's calculate these values.
Calculate the value by following steps:
- step0: Calculate:
  $$95-74-29+9$$
- step1: Calculate:
  $$1$$
Calculate or simplify the expression $$ 116 - 74 - 12 + 9 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$116-74-12+9$$
- step1: Calculate:
  $$39$$
Calculate or simplify the expression $$ 47 - 29 - 12 + 9 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$47-29-12+9$$
- step1: Calculate:
  $$15$$
Calculate or simplify the expression $$ 74 - 9 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$74-9$$
- step1: Subtract the numbers:
  $$65$$
Calculate or simplify the expression $$ 29 - 9 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$29-9$$
- step1: Subtract the numbers:
  $$20$$
Calculate or simplify the expression $$ 12 - 9 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$12-9$$
- step1: Subtract the numbers:
  $$3$$
Based on the calculations, we have the following counts:

1. Students with only Facebook: $$ F_{only} = 1 $$
2. Students with only Instagram: $$ I_{only} = 39 $$
3. Students with only Twitter: $$ T_{only} = 15 $$
4. Students with only Facebook and Instagram: $$ F \cap I_{only} = 65 $$
5. Students with only Facebook and Twitter: $$ F \cap T_{only} = 20 $$
6. Students with only Instagram and Twitter: $$ I \cap T_{only} = 3 $$
7. Students with all three accounts: $$ F \cap I \cap T = 9 $$
8. Students with no accounts: 33

Now, we can represent these values in a Venn diagram.
The Venn diagram will have three circles representing Facebook, Instagram, and Twitter. The counts will be placed in the respective sections:

- Only Facebook: 1
- Only Instagram: 39
- Only Twitter: 15
- Facebook and Instagram only: 65
- Facebook and Twitter only: 20
- Instagram and Twitter only: 3
- All three: 9
- Outside all circles (no accounts): 33

This arrangement will accurately represent the survey data. You can choose the diagram that matches this description.
The accurate Venn diagram will show the counts as described above, reflecting the distribution of social media accounts among the surveyed students.

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