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, 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 a) how many college students were surveyed? (simplify your answer.)

Explanation:

Step1: Use the principle of inclusion - exclusion

Let \(T\) be the set of students who own a tablet, \(L\) be the set of students who own a laptop, and \(G\) be the set of students who own a gaming system. We know that \(n(T) = 360\), \(n(L)=292\), \(n(G) = 283\), \(n(T\cap L)=195\), \(n(L\cap G)=137\), \(n(T\cap G)=68\), \(n(T\cap L\cap G)=24\), and the number of students who own none of the devices \(n((T\cup L\cup G)^c)=24\).
The formula for \(n(T\cup L\cup G)\) is \(n(T\cup L\cup G)=n(T)+n(L)+n(G)-n(T\cap L)-n(L\cap G)-n(T\cap G)+n(T\cap L\cap G)\).

Step2: Calculate \(n(T\cup L\cup G)\)

Substitute the values into the formula:
\[

$$\begin{align*} n(T\cup L\cup G)&=360 + 292+283-195 - 137-68+24\\ &=(360+292+283)+24-(195 + 137+68)\\ &=935+24 - 400\\ &=559 \end{align*}$$

\]

Step3: Find the total number of surveyed students

The total number of surveyed students \(N=n(T\cup L\cup G)+n((T\cup L\cup G)^c)\).
Since \(n(T\cup L\cup G) = 559\) and \(n((T\cup L\cup G)^c)=24\), then \(N=559 + 24=583\).

Answer:

583