Question

a survey of 112 high school students determined whether they used instagram, twitter, or facebook. the provided information was determined.
64 used instagram 60 used twitter
73 used facebook 31 used instagram and twitter
40 used instagram and facebook. 39 used twitter and facebook
20 had all three features.
complete parts a) through e)

a) how many of the students surveyed used only instagram?
(type a whole number.)

Explanation:

Step1: Define the formula for only Instagram

To find the number of students who used only Instagram, we use the principle of inclusion - exclusion for three sets. The number of students who used only Instagram is the total number of Instagram users minus those who used Instagram with Twitter, minus those who used Instagram with Facebook, plus those who used all three (because we subtracted them twice).
Let \( I \) be the set of Instagram users, \( T \) be the set of Twitter users, and \( F \) be the set of Facebook users. The formula for only \( I \) is \( n(\text{only } I)=n(I)-n(I\cap T)-n(I\cap F)+n(I\cap T\cap F) \)

Step2: Substitute the values

We know that \( n(I) = 64 \), \( n(I\cap T)=31 \), \( n(I\cap F) = 40 \), and \( n(I\cap T\cap F)=20 \)
Substitute these values into the formula:
\( n(\text{only } I)=64 - 31-40 + 20 \)
First, calculate \( 64-31 = 33 \)
Then, \( 33-40=- 7 \)
Then, \( -7 + 20=13 \)

Answer:

13