In a survey conduced on 800 students of a school , 250 students were found to like tea and 300 like coffee , 150 like both tea and coffee .Find how many students like neither tea nor coffee ?

Explanation

To solve the problem, we can use the principle of inclusion-exclusion to find the number of students who like neither tea nor coffee. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the given values
- Total number of students surveyed, \( n(S) = 800 \)
- Number of students who like tea, \( n(T) = 250 \)
- Number of students who like coffee, \( n(C) = 300 \)
- Number of students who like both tea and coffee, \( n(T \cap C) = 150 \)

### Step 2: Calculate the number of students who like only tea
To find the number of students who like only tea, we subtract the number of students who like both tea and coffee from the total number of tea drinkers:
\[ n(\text{only tea}) = n(T) - n(T \cap C) = 250 - 150 = 100 \]

### Step 3: Calculate the number of students who like only coffee
Similarly, to find the number of students who like only coffee, we subtract the number of students who like both from the total number of coffee drinkers:
\[ n(\text{only coffee}) = n(C) - n(T \cap C) = 300 - 150 = 150 \]

### Step 4: Calculate the total number of students who like either tea or coffee
Now, we can find the total number of students who like either tea or coffee by adding those who like only tea, those who like only coffee, and those who like both:
\[ n(T \cup C) = n(\text{only tea}) + n(\text{only coffee}) + n(T \cap C) = 100 + 150 + 150 = 400 \]

### Step 5: Calculate the number of students who like neither tea nor coffee
Finally, to find the number of students who like neither tea nor coffee, we subtract the total number of students who like either from the total number of students surveyed:
\[ n(\text{neither tea nor coffee}) = n(S) - n(T \cup C) = 800 - 400 = 400 \]

### Final Answer
The number of students who like neither tea nor coffee is **400**.

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