in a survey of college students, each of the following was found. of these students, 360 owned a tablet, 292…

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.)

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.)

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