Out of 15 students studying in a class, 7 are from Maharastra, 5 are from Karnataka and 3 are from Goa, Four students are to be selected at random. What are the chances that at least one is from Karnataka?

Total possible ways of selecting 4 students out of 15 students = ¹⁵C₄
= (15 x 14 x 13 x 12) / (1 x 2 x 3 x 4) = 1365
The no. of ways of selecting 4 students in which no students belongs to karnataka = ¹⁰C₄
∴ Number of ways of selecting atleast one student from karnataka = ¹⁵C₄ - ¹⁰C₄ = 1155.
∴ Required probability
= 1155 / 1365 = 77 / 91 = 11 / 13