Question

There are six teachers. Out of them, two teach physics, other two teach Chemistry and the rest two teach Mathematics. They have to stand in a row such that Physics, Chemistry and Mathematics teachers are always in a set. The number of ways in which they can do, is:

• A. 48
• B. 36
• C. 24
• D. 12

Answer 1

The Correct Option is A

To solve this problem, we need to determine the number of ways six teachers can be arranged in a row, given that teachers of the same subject should always be grouped together. Here's a step-by-step breakdown of the solution: 

1. Identify the Groups:
   • There are 2 Physics teachers, 2 Chemistry teachers, and 2 Mathematics teachers. These three groups need to be arranged such that teachers of the same subject are always together.
2. Arrange the Groups as Blocks:
   • Since there are three groups (Physics, Chemistry, Mathematics), they can be arranged in 3! (factorial) ways.
   • Therefore, the number of ways to arrange these groups is: \(3! = 3 \times 2 \times 1 = 6\).
3. Arrange Teachers Within Each Group:
   • Within each subject group, the teachers can be arranged among themselves in 2! (factorial) ways since each group has 2 teachers.
   • Therefore, for Physics teachers: \(2! = 2 \times 1 = 2\)
   • Similarly, for Chemistry and Mathematics teachers, each can be arranged in 2! ways which is also 2.
   • This means for all three groups, the total arrangement within the groups is: \(2! \times 2! \times 2! = 2 \times 2 \times 2 = 8\).
4. Total Arrangements:
   • To get the total number of arrangements, multiply the ways to arrange the groups as blocks by the ways to arrange the teachers within each group:
   • Total arrangements = \(3! \times (2! \times 2! \times 2!) = 6 \times 8 = 48\).
5. Conclusion:
   • The number of ways the teachers can stand in a row, respecting the subject groupings, is 48.

Therefore, the correct answer is 48, which matches the given correct answer option.