Question

a magazine surveyed subscribers, asking which of the following reality shows they watched on a regular basis: survivor, the voice, americas got talent. the results showed that 358 watched survivor, 295 watched the voice, 285 watched americas got talent, 194 watched survivor and the voice, 202 watched survivor and americas got talent, 139 watched the voice and americas got talent, 68 watched survivor, the voice, and americas got talent, and 27 watched none of these shows. complete parts a) through e) below.
a) how many subscribers were surveyed?
(simplify your answer)

Explanation:

Step1: Apply the principle of inclusion - exclusion for three sets.

Let \( S \) be the set of subscribers who watched Survivor, \( V \) be the set who watched The Voice, and \( T \) be the set who watched America's Got Talent. The formula for \( |S\cup V\cup T| \) is \( |S|+|V|+|T|-|S\cap V|-|S\cap T|-|V\cap T|+|S\cap V\cap T| \).
We know \( |S| = 358 \), \( |V| = 295 \), \( |T| = 285 \), \( |S\cap V| = 194 \), \( |S\cap T| = 202 \), \( |V\cap T| = 139 \), and \( |S\cap V\cap T| = 68 \).
Substitute these values into the formula:
\[

$$\begin{align*} |S\cup V\cup T|&=358 + 295+285-194 - 202-139 + 68\\ &=(358+295 + 285)+68-(194 + 202+139)\\ &=(938)+68-(535)\\ &=1006 - 535\\ &= 471 \end{align*}$$

\]

Step2: Add the number of subscribers who watched none of the shows.

The number of subscribers who watched none of the shows is 27. So the total number of subscribers surveyed is \( |S\cup V\cup T|+27 \).
\[
471+27 = 498
\]

Answer:

498