DIRECTIONS for questions: Read the information given below and answer the question that follows.

At InnovateX, six employees, Asha, Bunty, Chintu, Dolly, Eklavya, and Falguni, were split into two groups of three each: Elite led by Manager Kuku, and Novice led by Manager Lalu.

At the end of each quarter, Kuku and Lalu handed out ratings to all members in their respective groups. In each group, each employee received a distinct integer rating from 1 to 3.

The score for an employee at the end of a quarter is defined as their cumulative rating from the beginning of the year. At the end of each quarter the employee in Novice with the highest score was promoted to Elite, and the employee in Elite with the minimum score was demoted to Novice. If there was a tie in scores, the employee with a higher rating in the latest quarter was ranked higher.

1. Asha, Bunty, and Chintu were in Elite at the beginning of Quarter 1. All of them were in Novice at the beginning of Quarter 4.

2. Dolly and Falguni were the only employees who got the same rating across all the quarters.

3. The following is known about ratings given by Lalu:

• Bunty received a rating of 1 in Quarter 2.
• Asha and Dolly received ratings of 1 and 2, respectively, in Quarter 3.

For how many employees can the scores at the end of Quarter 3 be determined with certainty?

The correct answer is 4.

Let's deduce the positions and ratings of each employee quarter by quarter based on the rules and given statements.

Step 1: Determine the Promoted and Demoted Employees
Initially, Asha, Bunty, and Chintu (A, B, C) are in the Elite group, while Dolly, Eklavya, and Falguni (D, E, F) are in the Novice group. At the start of Quarter 4, A, B, and C are all in Novice. Since exactly one employee is demoted from Elite to Novice each quarter, A, B, and C must be the three employees demoted at the end of Q1, Q2, and Q3, in some order. Similarly, D, E, and F must be the three employees promoted to Elite at the end of Q1, Q2, and Q3, in some order.

Step 2: Analyze Constant Ratings
Dolly and Falguni received the same rating across all quarters. They both started in the Novice group. In Quarter 3, we know Dolly received a rating of 2 from Lalu (the Novice manager). This means Dolly was still in Novice in Q3, and her constant rating for all quarters is 2. Falguni's constant rating must be either 1 or 3.

Step 3: Track Novice Group and Promotions
In Q1 Novice: We have D, E, and F. Dolly gets 2. Falguni must get 3 (if F got 1, then F's rating in Q2 Novice would also be 1, but we know Bunty gets 1 in Q2 Novice, and ratings must be distinct). Thus, Falguni gets 3 and Eklavya gets 1. Falguni has the highest score ($3$) and is promoted to Elite at the end of Q1.

In Q2 Novice: The group has Dolly, Eklavya, and the employee demoted at the end of Q1. Since Bunty received a rating of 1 from Lalu in Q2, Bunty must be the one demoted in Q1. Ratings in Q2: Bunty gets 1, Dolly gets her constant 2. Therefore, Eklavya gets 3.
Scores at the end of Q2: Dolly = $2+2=4$, Eklavya = $1+3=4$, Bunty = $1$ (from Q1) $+1=2$. Since Eklavya and Dolly have the same score, Eklavya is promoted due to a higher rating in the latest quarter ($3>2$).

In Q3 Novice: The group has Dolly, Bunty, and the employee demoted in Q2. We are given that Asha receives 1 and Dolly receives 2 in Q3. Thus, Asha must be the one demoted in Q2. This leaves Bunty to receive the rating of 3. Dolly is promoted at the end of Q3 since she is the only original Novice member left to be promoted.

Step 4: Track Elite Group and Demotions
In Q1 Elite: The group has A, B, and C. Since B is demoted, B must have the lowest rating, which is 1. A and C receive 2 and 3 in some order.

In Q2 Elite: The group has A, C, and F (promoted in Q1). Falguni gets her constant rating of 3. Asha is the one demoted at the end of Q2. A and C must receive 1 and 2. For Asha to be demoted, she must receive 1, and Chintu receives 2. (If Asha received 2 and Chintu 1, A's score would be $2+2=4$ or $3+2=5$, and C's score would be $3+1=4$ or $2+1=3$, which would mean Chintu gets demoted, a contradiction).

In Q3 Elite: The group has C, F, and E (promoted in Q2). Falguni gets her constant 3. Chintu is the one demoted at the end of Q3. C and E must receive 1 and 2. Eklavya's score before Q3 is 4. Chintu's score before Q3 is either $3+2=5$ or $2+2=4$. For Chintu to be demoted, Chintu must receive 1, and Eklavya receives 2.

Step 5: Calculate Quarter 3 Scores
Now we calculate the cumulative scores at the end of Quarter 3 for all employees:

• Dolly: $2$ (Q1) + $2$ (Q2) + $2$ (Q3) = $6$
• Falguni: $3$ (Q1) + $3$ (Q2) + $3$ (Q3) = $9$
• Eklavya: $1$ (Q1) + $3$ (Q2) + $2$ (Q3) = $6$
• Bunty: $1$ (Q1) + $1$ (Q2) + $3$ (Q3) = $5$
• Asha: Q1 rating could be 2 or 3, plus $1$ (Q2) + $1$ (Q3). Score = $4$ or $5$.
• Chintu: Q1 rating could be 3 or 2, plus $2$ (Q2) + $1$ (Q3). Score = $6$ or $5$.

The scores can be determined with certainty for exactly 4 employees (Dolly, Falguni, Eklavya, and Bunty).