Question

in a survey of college students, each of the following was found. of these students, 360 owned a tablet, 292 owned a laptop, 283 owned a gaming system, 195 owned a tablet and a laptop, 202 owned a tablet and a gaming system, 137 owned a laptop and a gaming system, 68 owned a tablet, a laptop, and a gaming system, and 24 owned none of these devices. complete parts a) through c) below.
a) how many college students were surveyed?
493 (simplify your answer.)
b) of the college students surveyed, how many owned a tablet and a gaming system, but not a laptop?
134 (simplify your answer.)
c) of the college students surveyed, how many owned a laptop, but neither a tablet nor a gaming system?
(simplify your answer.)

Explanation:

Step1: Recall the principle of inclusion - exclusion

Let \(T\) be the set of tablet - owners, \(L\) be the set of laptop - owners, and \(G\) be the set of gaming - system owners. The formula for \(|T\cup L\cup G|\) is \(|T|+|L|+|G|-|T\cap L|-|T\cap G|-|L\cap G|+|T\cap L\cap G|\).
We know \(|T| = 360\), \(|L|=292\), \(|G| = 283\), \(|T\cap L|=195\), \(|T\cap G| = 202\), \(|L\cap G|=137\), and \(|T\cap L\cap G| = 68\).
First, calculate \(|T\cup L\cup G|\):
\[

$$\begin{align*} |T\cup L\cup G|&=360 + 292+283-195 - 202-137+68\\ &=(360+292+283)+68-(195 + 202+137)\\ &=935+68 - 534\\ &=469 \end{align*}$$

\]
Since 24 students own none of these devices, the total number of students surveyed is \(|T\cup L\cup G|+24=469 + 24=493\).

Step2: Find the number of students who own a tablet and a gaming system but not a laptop

The number of students who own a tablet and a gaming system is \(|T\cap G|\). The number of students who own a tablet, a gaming system, and a laptop is \(|T\cap L\cap G|\).
The number of students who own a tablet and a gaming system but not a laptop is \(|T\cap G|-|T\cap L\cap G|=202 - 68=134\).

Step3: Find the number of students who own a laptop but neither a tablet nor a gaming system

We use the principle of inclusion - exclusion. First, find the number of students in \(L-(T\cup G)\).
We know that \(|L\cap(T\cup G)|=|(L\cap T)\cup(L\cap G)|\). By the inclusion - exclusion principle, \(|(L\cap T)\cup(L\cap G)|=|L\cap T|+|L\cap G|-|L\cap T\cap G|=195 + 137-68=264\).
The number of students who own a laptop but neither a tablet nor a gaming system is \(|L|-|L\cap(T\cup G)|=292-264 = 28\).

Answer:

28