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 e) 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?
28 (simplify your answer.)
d) of the college students surveyed, how many owned exactly two of these devices?
330 (simplify your answer.)
e) of the college students surveyed, how many owned at least one of these devices?
□ (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 number of students who own at least one device is \(n(T\cup L\cup G)=n(T)+n(L)+n(G)-n(T\cap L)-n(T\cap G)-n(L\cap G)+n(T\cap L\cap G)\).
We know that \(n(T) = 360\), \(n(L)=292\), \(n(G)=283\), \(n(T\cap L)=195\), \(n(T\cap G)=202\), \(n(L\cap G)=137\), and \(n(T\cap L\cap G)=68\).

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

\[

$$\begin{align*} n(T\cup L\cup G)&=360 + 292+283-195 - 202-137+68\\ &=(360 + 292+283)+68-(195 + 202+137)\\ &=935+68 - 534\\ &=1003 - 534\\ &=469 \end{align*}$$

\]

Step3: Find the number of students who own at least one device

The number of students who own at least one device is the number of students in \(T\cup L\cup G\). We know that 24 students own none of the devices.
The total number of students surveyed is the number of students who own at least one device plus the number of students who own none of the devices. So the number of students who own at least one device is \(469\).

Answer:

469