Explanation: To solve this problem, we need to calculate the probability that at least one of the four selected students is from Karnataka. This can be approached using the concept of complementary probability, which involves calculating the probability of the opposite event (no students from Karnataka) and subtracting it from 1.
First, let's determine the total number of ways to select 4 students out of 15. This is given by the combination formula \( \binom{n}{k} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose. Here, \( n = 15 \) and \( k = 4 \):
\[
\binom{15}{4} = \frac{15!}{4!(15-4)!} = \frac{15!}{4! \cdot 11!} = \frac{15 \times 14 \times 13 \times 12}{4 \times 3 \times 2 \times 1} = 1365
\]
Next, we calculate the number of ways to select 4 students such that none of them are from Karnataka. Since there are 5 students from Karnataka, there are \( 15 - 5 = 10 \) students who are not from Karnataka. We need to choose 4 students from these 10:
\[
\binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4! \cdot 6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\]
The probability of selecting 4 students such that none of them are from Karnataka is the ratio of the number of favorable outcomes to the total number of outcomes:
\[
P(\text{no students from Karnataka}) = \frac{\binom{10}{4}}{\binom{15}{4}} = \frac{210}{1365} = \frac{2}{13}
\]
The probability of at least one student from Karnataka being selected is the complement of the above probability:
\[
P(\text{at least one student from Karnataka}) = 1 - P(\text{no students from Karnataka}) = 1 - \frac{2}{13} = \frac{13}{13} - \frac{2}{13} = \frac{11}{13}
\]
Thus, the probability that at least one of the four selected students is from Karnataka is \( \frac{11}{13} \), which corresponds to option C.
This problem demonstrates the use of complementary probability and combinatorial calculations to solve probability problems involving selection without replacement. Understanding these concepts is crucial for solving similar problems in competitive exams and other probability-related questions.