Question

construct a venn diagram illustrating these results. then answer the question.

• 699 incoming freshmen were surveyed in all
• 322 of the freshmen like classical
• 375 of the freshmen like rap
• 205 of the freshmen like both rap and classical
• 270 of the freshmen like both rap and rock
• 255 of the freshmen like both classical and rock
• 163 of the freshmen like all three types of music (rap, classical, rock)
• 123 of the freshmen don’t like any of these three types of music (rap, classical, rock)

how many of the freshmen like rap, but like neither classical nor rock?

Explanation:

Step1: Define the sets

Let \( R \) be the set of freshmen who like rap, \( C \) be the set who like classical, and \( K \) be the set who like rock. We know the total number of freshmen \( n(\text{Total}) = 699 \), \( n(R) = 322 \), \( n(C) = 205 \) (wait, no, wait: Wait, the problem says: 322 like rap? Wait no, let's re-express the data:

Wait the data given:

- 699 incoming freshmen surveyed.

- 322 like rap? Wait no, let's parse the text:

Wait the bullet points:

- 322 of the freshmen like rap? Wait no, let's list all:

- 322 of the freshmen like rap? Wait no, let's read again:

Wait the text:

- 699 incoming freshmen were surveyed in all

- 322 of the freshmen like rap? Wait no, the first bullet after 699: "322 of the freshmen like rap"? Wait no, let's list all:

Wait the bullet points (from the image text):

- 699 incoming freshmen were surveyed in all

- 322 of the freshmen like rap? Wait no, let's parse each:

Wait the text:

"• 699 incoming freshmen were surveyed in all

• 322 of the freshmen like rap

• 375 of the freshmen like classical

• 205 of the freshmen like both rap and classical

• 270 of the freshmen like both rap and rock

• 255 of the freshmen like both classical and rock

• 163 of the freshmen like all three types of music (rap, classical, rock)

• 123 of the freshmen don't like any of these three types of music (rap, classical, rock)"

Wait, no, wait: Wait the original text (from the image):

Wait the user's image text:

"699 incoming freshmen were surveyed in all

322 of the freshmen like rap

375 of the freshmen like classical

205 of the freshmen like both rap and classical

270 of the freshmen like both rap and rock

255 of the freshmen like both classical and rock

163 of the freshmen like all three types of music (rap, classical, rock)

123 of the freshmen don't like any of these three types of music (rap, classical, rock)"

Wait, but we need to find the number of freshmen who like rap, but neither classical nor rock. That is, \( n(R \cap \overline{C} \cap \overline{K}) \).

To find this, we use the principle of inclusion - exclusion for three sets. The formula for \( n(R) \) is:

\( n(R) = n(R \cap \overline{C} \cap \overline{K}) + n(R \cap C \cap \overline{K}) + n(R \cap \overline{C} \cap K) + n(R \cap C \cap K) \)

We know:

- \( n(R \cap C \cap K) = 163 \) (like all three)

- \( n(R \cap C) = 205 \) (both rap and classical). But \( n(R \cap C) = n(R \cap C \cap \overline{K}) + n(R \cap C \cap K) \). So \( n(R \cap C \cap \overline{K}) = n(R \cap C) - n(R \cap C \cap K) = 205 - 163 = 42 \)

- \( n(R \cap K) = 270 \) (both rap and rock). Similarly, \( n(R \cap K) = n(R \cap \overline{C} \cap K) + n(R \cap C \cap K) \). So \( n(R \cap \overline{C} \cap K) = 270 - 163 = 107 \)

- \( n(R) = 322 \) (total who like rap). So:

\( n(R) = n(R \cap \overline{C} \cap \overline{K}) + n(R \cap C \cap \overline{K}) + n(R \cap \overline{C} \cap K) + n(R \cap C \cap K) \)

Plug in the known values:

\( 322 = n(R \cap \overline{C} \cap \overline{K}) + 42 + 107 + 163 \)

Calculate the sum of the known terms: \( 42 + 107 + 163 = 312 \)

So \( n(R \cap \overline{C} \cap \overline{K}) = 322 - 312 = 10 \)

Wait, but the question is: "How many of the freshmen like rap, but neither classical nor rock?"

So that's \( n(R \cap \overline{C} \cap \overline{K}) \), which we calculated as 10? Wait, but let's verify with the total.

Wait, the total number of freshmen who like at least one music type is \( 699 - 123 = 576 \) (since 123 don't like any).

Let's check the inclusion - exclusion principle for three sets:

\( n(R \cup C \cup K) = n(R) + n(C) + n(K) - n(R \cap C) - n(R \cap K) - n(C \cap K) + n(R \cap C \cap K) \)

We know \( n(R) = 322 \), \( n(C) = 375 \), \( n(K) \): Wait, we don't know \( n(K) \) (like rock) directly. Wait, but we can find \( n(K) \) using the data. Wait, but maybe we don't need it. Wait, but let's check our calculation for \( n(R \cap \overline{C} \cap \overline{K}) \).

Wait, \( n(R) = \) only rap + rap and classical only + rap and rock only + all three.

Only rap: \( x \)

Rap and classical only: \( n(R \cap C) - n(R \cap C \cap K) = 205 - 163 = 42 \)

Rap and rock only: \( n(R \cap K) - n(R \cap C \cap K) = 270 - 163 = 107 \)

All three: 163

So \( x + 42 + 107 + 163 = 322 \)

\( x + 312 = 322 \)

\( x = 10 \)

Yes, that's correct.

But wait, the question in the image is: "How many of the freshmen like rap, but neither classical nor rock?"

So that's \( x = 10 \)? Wait, but let's confirm with the total.

Total who like at least one: \( 699 - 123 = 576 \)

Now, let's calculate \( n(R \cup C \cup K) \) using inclusion - exclusion:

\( n(R) = 322 \), \( n(C) = 375 \), \( n(K) \): Let's find \( n(K) \).

\( n(K) = \) only rock + rock and classical only + rock and rap only + all three.

Rock and classical only: \( n(C \cap K) - n(R \cap C \cap K) = 255 - 163 = 92 \)

Rock and rap only: 107 (as above)

All three: 163

Only rock: \( y \)

So \( n(K) = y + 92 + 107 + 163 \)

But also, \( n(C \cap K) = 255 \) (both classical and rock) = rock and classical only + all three = 92 + 163 = 255, which checks out.

Now, \( n(R \cup C \cup K) = n(R) + n(C) + n(K) - n(R \cap C) - n(R \cap K) - n(C \cap K) + n(R \cap C \cap K) \)

We know \( n(R \cup C \cup K) = 576 \)

So \( 576 = 322 + 375 + n(K) - 205 - 270 - 255 + 163 \)

Calculate the right - hand side:

\( 322+375 = 697 \); \( 697 + n(K)=697 + n(K) \); \( 697 + n(K)-205 = 492 + n(K) \); \( 492 + n(K)-270 = 222 + n(K) \); \( 222 + n(K)-255 = n(K)-33 \); \( n(K)-33 + 163 = n(K)+130 \)

So \( 576 = n(K)+130 \)

Thus, \( n(K)=576 - 130 = 446 \)

Now, check \( n(K) \) using the components:

Only rock \( y \), rock and classical only 92, rock and rap only 107, all three 163. So \( y + 92 + 107 + 163 = y + 362 = 446 \), so \( y = 446 - 362 = 84 \). Which is consistent, but not necessary for our problem.

Back to the question: "How many like rap, but neither classical nor rock?"

We found that as \( x = 10 \). Wait, but let's confirm with the total who like rap:

Only rap (x) + rap and classical only (42) + rap and rock only (107) + all three (163) = \( x + 42 + 107 + 163 = x + 312 \). And \( n(R)=322 \), so \( x = 322 - 312 = 10 \).

Yes, that's correct.

Wait, but let's check the total number of freshmen:

Number who don't like any: 123

Number who like at least one: 699 - 123 = 576

Now, let's calculate \( n(R \cup C \cup K) \) using the components:

Only rap: 10

Only classical: Let's find it. \( n(C) = 375 \). Only classical \( z \), classical and rap only 42, classical and rock only 92, all three 163. So \( z + 42 + 92 + 163 = z + 297 = 375 \), so \( z = 375 - 297 = 78 \)

Only rock: 84 (as above)

Rap and classical only: 42

Rap and rock only: 107

Classical and rock only: 92

All three: 163

Now, sum all these: 10 (only rap) + 78 (only classical) + 84 (only rock) + 42 (rap - classical) + 107 (rap - rock) + 92 (classical - rock) + 163 (all three) =

10 + 78 = 88; 88 + 84 = 172; 172 + 42 = 214; 214 + 107 = 321; 321 + 92 = 413; 413 + 163 = 576. Which matches 699 - 123 = 576. So the components are correct.

Thus, the number of freshmen who like rap, but neither classical nor rock is 10.

Wait, but the question in the image (the lower part) is: "How many of the freshmen like rap, but neither classical nor rock?"

Yes, so the answer is 10.

Wait, but let's re - check the data:

Wait, the problem says "like rap, but neither classical nor rock" – that is, rap only, not in classical or rock.

So using the formula for the size of a set in a Venn diagram with three circles:

For set \( R \) (rap), the number of elements only in \( R \) is \( n(R) - n(R \cap C) - n(R \cap K) + n(R \cap C \cap K) \) (by the principle of inclusion - exclusion for the intersection parts). Wait, the formula for only \( R \) is \( n(R) - [n(R \cap C) + n(R \cap K) - n(R \cap C \cap K)] \)

Because \( n(R \cap C) + n(R \cap K) \) counts \( n(R \cap C \cap K) \) twice, so we subtract it once to get the number of elements in \( R \) and either \( C \) or \( K \) (or both). Then, subtracting that from \( n(R) \) gives the number of elements only in \( R \).

So:

Only \( R = n(R) - [n(R \cap C) + n(R \cap K) - n(R \cap C \cap K)] \)

Plugging in the values:

\( n(R) = 322 \), \( n(R \cap C)=205 \), \( n(R \cap K)=270 \), \( n(R \cap C \cap K)=163 \)

So:

Only \( R = 322 - (205 + 270 - 163) \)

Calculate inside the parentheses: \( 205 + 270 = 475 \); \( 475 - 163 = 312 \)

Then, \( 322 - 312 = 10 \)

Yes, that's the same result. So the number of freshmen who like rap, but neither classical nor rock is 10.

Answer:

\( \boxed{10} \)