We are given a set of 144 students numbered from 1 to 144, where students are categorized based on their preparation for different fields:

1. MBA: Students with even numbers.
2. UPSC: Students whose numbers are divisible by 3.
3. Law: Students whose numbers are divisible by 5.
4. BBA: All other students.

Approach:

1. Total number of students: 144.
2. Find the sets of students in different categories, while ensuring that students who fit into multiple categories (like numbers divisible by both 2 and 3, etc.) are counted once.

Step 1: Students preparing for MBA (even-numbered)

The students preparing for MBA are the even-numbered students. These are the multiples of 2 between 1 and 144:

• Numbers: 2, 4, 6, 8, ..., 144.
• The total count is given by the sequence $2n = 2, 4, 6, ..., 144$, where $n = 72$.

Thus, 72 students are preparing for MBA.

Step 2: Students preparing for UPSC (divisible by 3)

The students preparing for UPSC are the multiples of 3 between 1 and 144:

• Numbers: 3, 6, 9, 12, ..., 144.
• The total count is $\frac{144}{3} = 48$ students preparing for UPSC.

Step 3: Students preparing for Law (divisible by 5)

The students preparing for Law are the multiples of 5 between 1 and 144:

• Numbers: 5, 10, 15, ..., 140.
• The total count is $\frac{144}{5} = 28$ students preparing for Law.

Step 4: Using the Inclusion-Exclusion Principle

There is some overlap among these groups, so we need to account for students counted in multiple categories:

• MBA ∩ UPSC (divisible by 6): These are multiples of 6. There are $\frac{144}{6} = 24$ such students.
• MBA ∩ Law (divisible by 10): These are multiples of 10. There are $\frac{144}{10} = 14$ such students.
• UPSC ∩ Law (divisible by 15): These are multiples of 15. There are $\frac{144}{15} = 9$ such students.
• MBA ∩ UPSC ∩ Law (divisible by 30): These are multiples of 30. There are $\frac{144}{30} = 4$ such students.

Step 5: Total number of students in each group using inclusion-exclusion

Now, applying the Inclusion-Exclusion Principle to find the total number of students who are preparing for MBA, UPSC, or Law:

$|MBA \cup UPSC \cup Law| = (72 + 48 + 28) - (24 + 14 + 9) + 4 = 148 - 47 + 4 = 105$

So, 105 students are preparing for MBA, UPSC, or Law.

Step 6: Students preparing for BBA

The students preparing for BBA are the ones not included in any of these categories:

$\text{Students preparing for BBA} = 144 - 105 = 39$

Final Answer:

39 students are preparing for BBA.

---

Would you like a more detailed breakdown or have any questions about this solution?

Here are 5 related questions to explore further:

1. What is the general formula for finding the number of multiples of a number within a range?
2. How does the inclusion-exclusion principle work for more than three sets?
3. How would the solution change if the range of students was different, say from 1 to 200?
4. How would we modify the approach if some students could prepare for more than one field?
5. Can this problem be represented using Venn diagrams, and how would that visualization look?

Tip: Always remember to account for overlaps when working with multiple conditions using the inclusion-exclusion principle.