The number of student passing or failing in an exam for a particular subject are presented in the bar chart below. Students who pass the exam cannot appear for the exam again. Students who fails the exam in the first attempt must appear for the exam in the following year. Students always pass the exam in their second attempt. The numbers of students who took the exam for the first time in the year 2nd the year 3 respectively, are ________.

The correct option is 55 and 48.

To find the number of students taking the exam for the first time in Year 2 and Year 3, we analyze the data provided in the bar chart and the rules of the exam:

Data from the Bar Chart:
- Year 1: 50 students passed (represented by the light gray bar labeled 50) and 10 students failed (represented by the dark bar labeled 10).
- Year 2: 60 students passed (represented by the light gray bar labeled 60) and 5 students failed (represented by the dark bar labeled 5).
- Year 3: 50 students passed (represented by the light gray bar labeled 50) and 3 students failed (represented by the dark bar labeled 3).

Rules:
1. Students who fail in their first attempt must reappear in the next year.
2. Students in their second attempt always pass. Therefore, all failing students in any year must be taking the exam for the first time.

Step-by-Step Analysis:

1. Calculation for Year 2:
The 10 students who failed in Year 1 must appear for the exam in Year 2. Since it is their second attempt, they are guaranteed to pass. Thus, of the 60 passing students in Year 2, 10 are second-time takers.
The number of first-time takers who passed in Year 2 is:
$60-10=50$
The 5 students who failed in Year 2 must be first-time takers. Therefore, the total number of students who took the exam for the first time in Year 2 is:
$50+5=55$

2. Calculation for Year 3:
The 5 students who failed in Year 2 must appear in Year 3, and all of them pass (second attempt). Thus, of the 50 passing students in Year 3, 5 are second-time takers.
The number of first-time takers who passed in Year 3 is:
$50-5=45$
The 3 students who failed in Year 3 must be first-time takers. Therefore, the total number of students who took the exam for the first time in Year 3 is:
$45+3=48$

Thus, the number of first-time exam takers in Year 2 and Year 3 are 55 and 48, respectively.