Question

During the placement season of a class, 21 students got shortlisted for company A, 26 got shortlisted for Company B and 29 got shortlisted for company C and 14 students got shortlisted for both A and B,12 students got shortlisted for A and C and 15 for both B and C. All the companies shortlisted 8 students from the class. Then what is the ratio of number of students who got shortlisted for only B and number of students who got shortlisted for only C?

• A. 1:1
• B. 1:2
• C. 2:3
• D. 3:2

Answer 1

The Correct Option is B

Let the number of students who got shortlisted for all three companies be \( x \).
Using the principle of inclusion and exclusion for the total number of students shortlisted:
\[ 21 + 26 + 29 - 14 - 12 - 15 + x = \text{Total students shortlisted} \]
\[ 21 + 26 + 29 - 14 - 12 - 15 + x = 8 \]
\[ 35 + x = 8 \]
\[ x = 8 - 35 \]
\[ x = -27 \]
There seems to be a mistake in the total count or problem constraints. Based on the final expected total, let's reconsider without exclusions affecting directly.
To find the number of students shortlisted only for Company B and Company C, let's denote:
- Only B: \( B - (A \cap B) - (B \cap C) + (A \cap B \cap C) \)
- Only C: \( C - (A \cap C) - (B \cap C) + (A \cap B \cap C) \)
From the given:
- Shortlisted for B only = \( 26 - 14 - 15 + 8 = 5 \)
- Shortlisted for C only = \( 29 - 12 - 15 + 8 = 10 \)
Thus the ratio is:
\[ \frac{5}{10} = 1:2 \]
Answer: B 1:2

Answer 2

First find how many students were shortlisted for exactly two companies (not all three), by removing the triple-overlap from each pairwise overlap.

Exactly A and B only: \(14-8=6\)
Exactly A and C only: \(12-8=4\)
Exactly B and C only: \(15-8=7\)

Now subtract these exactly-two counts and the all-three count (8) from each company total to get the only-this-company counts:
Only B \(=26-6-7-8=5\)
Only C \(=29-4-7-8=10\)

1. Option A (1:1): Would require Only B = Only C, but \(5\neq10\).
2. Option B (1:2): \(5:10=1:2\), which matches exactly.
3. Option C (2:3): Does not match \(5:10\).
4. Option D (3:2): Does not match \(5:10\).

So the ratio of students shortlisted only for B to only for C is \(5:10=1:2\).

Hence, the correct answer is Option B: 1:2.