Question

When another survey was conducted in June among the same students, the result was different. All of them liked Coke. 12 liked Tea, but no one liked Coffee. How many students liked only Coke?

• A. 40
• B. 23
• C. 20
• D. 11
• A. No one liked Coke and Coffee.
• B. Some liked Coke and Tea.
• C. Number of students who liked both Tea and Coffee is less than 20.
• D. Only 10 liked both Tea and Coffee.

Answer 1

The Correct Option is B

Step 1: Read the diagram and find the total number of students.
The Venn diagram splits the class into seven groups based on which drinks each student liked in February. Adding every region in the diagram gives \(4+2+11+5+3+4+6=35\) students in the class.

Step 2: Set up the June survey conditions.
The June survey is a fresh one, so we only use the new facts given for it: every one of the 35 students liked Coke, 12 students liked Tea, and no one liked Coffee.
Since no one liked Coffee this time, each student falls into exactly one of two groups, either liking Coke and Tea both, or liking Coke only.

Step 3: Find the number who liked Coke only.
Every student likes Coke, so the 12 students who like Tea must also be counted inside the Coke group, since a student cannot like Tea without also liking Coke when all 35 students already like Coke.
So the students who like both Tea and Coke number 12, and the rest of the class likes Coke only.
\[ \text{Only Coke} = 35 - 12 = 23 \]

Step 4: Check the other options.
40 is more than the total class size of 35, so it cannot be right.
20 and 11 would leave 15 or 24 students as Tea drinkers, which does not match the given 12.

Final Answer:
23 students liked only Coke.
\[ \boxed{23} \]

Answer 2

Step 1: List what the December survey tells us.
The class still has 35 students, the same class size as February. This time every student liked at least one hot drink, that is, every student liked Tea or Coffee or both. 16 students liked Coke, and no student liked all three drinks together.

Step 2: Work out how the 16 Coke drinkers are split.
Since every student, including the 16 who like Coke, likes at least one hot drink, none of the 16 Coke drinkers can be a Coke only student. Since no one likes all three drinks, none of the 16 can like Tea and Coffee at the same time as Coke.
So each of the 16 Coke drinkers likes exactly one hot drink along with Coke, either Tea and Coke, or Coffee and Coke. Adding these two groups gives exactly 16 students.

Step 3: Work out the remaining students.
The other \(35-16=19\) students do not like Coke at all, so between them they can only like Tea only, Coffee only, or both Tea and Coffee, since no one likes all three and none of them like Coke. These 19 students split across three groups: Tea only, Coffee only, and Tea and Coffee both.
The group who liked both Tea and Coffee, with no Coke since the all three group is empty, is only a part of these 19 students, so it can be at most 19, and only if every single one of them liked both drinks.

Step 4: Check each option against this reasoning.
Option 1, no one liked Coke and Coffee, is not forced, since the 16 Coke drinkers could split any way between Tea and Coke, and Coffee and Coke, including some who like Coffee and Coke.
Option 2, some liked Coke and Tea, is also not forced, since all 16 Coke drinkers could instead like Coffee and Coke, leaving zero for Tea and Coke.
Option 4, only 10 liked both Tea and Coffee, picks one exact number out of a whole range of possibilities from 0 to 19, so it is not guaranteed.
Option 3 says the number who liked both Tea and Coffee is less than 20. Since this number can be at most 19, as shown in Step 3, it is always less than 20, no matter how the 19 non Coke students split up.

Final Answer:
The number of students who liked both Tea and Coffee is always less than 20, so option 3 is the conclusion that must be true.
\[ \boxed{\text{Option 3}} \]