An investigator interviewed 100 students to determine their preferences for the three drinks : milk

Q: Why do we subtract the 10 from the 20, 30, and 25? Isn't that double counting?

Think of the 20 students who drink Milk and Coffee as a group that includes the 10 who drink all three. If you don't subtract them, you are counting those 10 students twice—once in the 'all three' group and once in the 'Milk and Coffee' group. By subtracting, we isolate the students who drink *only* Milk and Coffee.

Q: What if the problem didn't say 'only' for the single drink categories? How would that change things?

If the problem said '12 students had Milk' instead of '12 students had Milk only', we would have to subtract the intersections from that 12 to find the 'Milk only' count. The word 'only' is a gift—it saves us from having to perform that extra subtraction step.

Q: Is there a formula for this, or do I have to draw the diagram every time?

There is the Principle of Inclusion-Exclusion: $n(M \cup C \cup T) = n(M) + n(C) + n(T) - n(M \cap C) - n(C \cap T) - n(M \cap T) + n(M \cap C \cap T)$. However, for complex problems with many regions, the Venn diagram is much safer and more intuitive. It prevents algebraic errors.

Question

An investigator interviewed 100 students to determine their preferences for the three drinks : milk ($M$), coffee ($C$) and tea ($T$). He reported the following : 10 students had all the three drinks $M, C$ and $T$; 20 had $M$ and $C$; 30 had $C$ and $T$; 25 had $M$ and $T$; 12 had $M$ only; 5 had $C$ only; and 8 had $T$ only. Using a Venn diagram find how many did not take any of the three drinks.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are a detective tasked with solving a mystery. You have a group of $100$ students, and you need to figure out their drink preferences: Milk ($M$), Coffee ($C$), and Tea ($T$).

The universal set is defined as $n(U) = 100$, representing every student in the study. We use a Venn diagram to partition this territory into distinct, non-overlapping regions.

### The Heart of the Mystery: The Triple Intersection

The most common mistake is starting from the outside. Always start from the very center—the heart of the diagram.

This is the region where all three circles overlap, representing the students who enjoy all three drinks:
$$n(M \cap C \cap T) = 10$$

By placing this $10$ in the center, we anchor our entire diagram. Every other calculation is performed relative to this central value.

### The 'Only' Trap: Unmasking the Overlaps

Next, we look at the intersections of two sets. The problem states that $20$ students had Milk and Coffee.

This $20$ includes the $10$ students who had all three. To find the students who had *only* Milk and Coffee, we perform a subtraction:
$$20 - 10 = 10$$

We apply this same logical rigor to the other intersections:
*   For Coffee and Tea, the total is $30$, so the 'only' region is $30 - 10 = 20$.
*   For Milk and Tea, the total is $25$, so the 'only' region is $25 - 10 = 15$.

By subtracting the central $10$, we successfully isolate the students who enjoy exactly two drinks.

### Filling the Outer Shells

Finally, we address the students who stick to just one drink. The problem explicitly states: $12$ for Milk only, $5$ for Coffee only, and $8$ for Tea only.

Because the word 'only' is used, we do not need to subtract anything. We simply place these values in the outer, non-overlapping parts of our circles.

### The Grand Summation: Finding the Union

To find out how many students had at least one drink, we sum all these disjoint regions:
$$n(M \cup C \cup T) = 12 + 5 + 8 + 10 + 20 + 15 + 10 = 80$$

These $80$ students represent everyone who consumes at least one of the three drinks. To find the students who did not take any of the three drinks, we subtract the union from the universal set:
$$n(M \cup C \cup T)^c = n(U) - n(M \cup C \cup T)$$
$$n(M \cup C \cup T)^c = 100 - 80 = 20$$

**Final Answer:** There are **$20$** students who did not take any of the three drinks.

Enter Numerical Value:

Visualized Solution

Universal Set and Categories

The Triple Intersection $n (M \cap C \cap T)$

Calculating $n (M \cap C only)$

Calculating $n (C \cap T only)$

Calculating $n (M \cap T only)$

Identifying 'Only' Regions

Union of Sets $n (M \cup C \cup T)$

Students Taking No Drinks

The Sigma Insight: Types of Sets and Set Operations

Analyzing the Setup

The Heart of the Mystery

The 'Only' Trap

Filling the Outer Shells

The Grand Summation

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