In a class of 100 students, 73 like coffee, 80 like tea, and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then, the difference between the maximum and minimum possible number of students who like all the three drinks is:

Question

Accepted Answer

The problem involves finding the difference between the maximum and minimum possible number of students who like all three drinks—coffee, tea, and lemonade—in a class of 100 students.

### Given:
- Total students = 100
- Students who like coffee (C) = 73
- Students who like tea (T) = 80
- Students who like lemonade (L) = 52
- Some students may not like any of the three drinks.

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### Step 1: Define variables

- $$ n $$ = number of students who like none of the drinks
- $$ s $$ = number of students who like exactly one drink
- $$ d $$ = number of students who like exactly two drinks
- $$ t $$ = number of students who like all three drinks

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### Step 2: Form equations

1. Total students:
$$
n + s + d + t = 100
$$

2. Total "likes" counted by summing individual drink preferences:
$$
s + 2d + 3t = 73 + 80 + 52 = 205
$$

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### Step 3: Derive relation between $$d, t, n$$

Subtract equation 1 from equation 2:
$$
(s + 2d + 3t) - (n + s + d + t) = 205 - 100
$$
$$
d + 2t - n = 105
$$

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### Step 4: Find maximum and minimum values of $$ t $$

- **Maximum $$ t $$:**

Since lemonade has the smallest group (52), the maximum number of students who like all three drinks cannot exceed 52.

Set $$ t = 52 $$, then:
$$
d + 2(52) - n = 105 \implies d - n = 105 - 104 = 1
$$

Choose $$ n = 0 $$ (no one dislikes all drinks), so $$ d = 1 $$.

- **Minimum $$ t $$:**

Set $$ n = 0 $$ again for simplicity.

Then:
$$
d + 2t = 105
$$

To minimize $$ t $$, maximize $$ d $$.

Set $$ t = 5 $$, then:
$$
d + 2(5) = 105 \implies d = 105 - 10 = 95
$$

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### Step 5: Calculate the difference

$$
	ext{Difference} = 	ext{Maximum } t - 	ext{Minimum } t = 52 - 5 = 47
$$

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## Final Answer:
The difference between the maximum and minimum possible number of students who like all three drinks is **47**.

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