A college has 10 (non-overlapping) time slots for its courses, and blithely assigns courses to time slots randomly and independently. A student randomly chooses 3 of the courses to enroll in (for the PTP, to avoid getting fined). What is the probability that there is a conflict in the student’s schedule?
The solution provided uses complement to solve this question, what is wrong with my solution?
Probability of conflict is complement of not having conflict. Which is $$\frac{10\times 9 \times 8}{10^3}=0.72$$ The way you have done it is a little off. Ways of having two confilcts is you can have 3 ways in which the courses having conflict can be selected. The time slot of one of the courses can be chosen in 10 ways and the next two courses in 9 ways. So that is $3\times9\times10=270.$ Now 3 conflicts can be chosen in $10$ ways. Because once you choose the first course all the courses should be in the same slot. So that total is $280$ ways. So the probability of clash is $0.28$.