We will solve this problem using the principle of inclusion-exclusion.

Information:

• Let the sets represent the people who listen to each genre:
  • $R$: the set of people who listen to rap.
  • $H$: the set of people who listen to heavy metal.
  • $A$: the set of people who listen to alternative rock.

Given values:

• $|R \cap H \cap A| = 7$ (listened to rap, heavy metal, and alternative rock)
• $|R \cap H| = 10$ (listened to rap and heavy metal)
• $|H \cap A| = 13$ (listened to heavy metal and alternative rock)
• $|R \cap A| = 12$ (listened to rap and alternative rock)
• $|R| = 17$ (listened to rap)
• $|H| = 24$ (listened to heavy metal)
• $|A| = 22$ (listened to alternative rock)
• 9 listened to none of these three channels.

Let’s calculate step-by-step.

(a) How many people were surveyed?

Using the principle of inclusion-exclusion, the total number of people surveyed who listened to at least one genre (let’s call it $n$) can be found by:

$n = |R \cup H \cup A| = |R| + |H| + |A| - |R \cap H| - |H \cap A| - |R \cap A| + |R \cap H \cap A|$

Substitute the values:

$n = 17 + 24 + 22 - 10 - 13 - 12 + 7$

$n = 35$

So, 35 people listened to at least one of the three genres.

Since 9 people listened to none of these channels, the total number of people surveyed is:

$35 + 9 = 44$

Thus, 44 people were surveyed.

(b) How many people listened to either rap or alternative rock?

To find the number of people who listened to either rap or alternative rock, we can use:

$|R \cup A| = |R| + |A| - |R \cap A|$

Substitute the values:

$|R \cup A| = 17 + 22 - 12 = 27$

So, 27 people listened to either rap or alternative rock.

(c) How many listened to heavy metal only?

To find how many listened to heavy metal only, we calculate the number of people who listened to heavy metal but not to rap or alternative rock. This is:

$|H \text{ only}| = |H| - (|R \cap H| + |H \cap A| - |R \cap H \cap A|)$

Substitute the values:

$|H \text{ only}| = 24 - (10 + 13 - 7) = 24 - 16 = 8$

So, 8 people listened to heavy metal only.

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Summary of answers:

• (a) 44 people were surveyed.
• (b) 27 people listened to either rap or alternative rock.
• (c) 8 people listened to heavy metal only.

Let me know if you need further clarification or have any questions!

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Further Exploration:

1. How would the inclusion-exclusion formula change if there were more genres involved?
2. How can you calculate the number of people who listened to exactly two genres (but not all three)?
3. What happens to the total surveyed number if 12 people listened to none of the three genres instead of 9?
4. How would the problem change if it were asking for the number of people who listened to all three genres exclusively?
5. What is the probability that a person surveyed listened to at least one genre?

Tip: When working with inclusion-exclusion problems, always start by organizing the information clearly and define your sets and their intersections. This will make it easier to apply the formula.