1) Exactly 5 days are selected from each of the 12 months.
a) (30 C 5)^2 / 360 C 60 - This is the probability of selecting exactly 5 days from one month squared, divided by the total number of possible sets of 60 days. Incorrect.
b) (30 C 5) / 360 C 60 - This is the probability of selecting exactly 5 days from one month, divided by the total number of possible sets of 60 days. Incorrect.
c) 30^12 / 360^60 - This is the probability of selecting any 60 days out of the year, without any restrictions. Incorrect.
d) 360 / (30 C 5)^12 - This is the probability of selecting any 60 days out of the year, divided by the total number of possible sets of 5 days from each month, raised to the power of 12 (since it happens in all 12 months). Incorrect.
Answer: None of the options are correct.
2) None of the selected days is from January.
a) 1 - (30 C 12)/ (360 C 60) - This is the probability of selecting any 60 days out of the year, minus the probability of selecting 12 days from January and any 48 days from the rest of the year, divided by the total number of all possible sets of 60 days. Incorrect.
b) 1 - (330 C 60)/(360 C 60) - This is the probability of selecting any 60 days out of the year, minus the probability of selecting 60 days from January and any 300 days from the rest of the year, divided by the total number of all possible sets of 60 days. Incorrect.
c) (330 C 60) / (360 C 60) - This is the probability of selecting 60 days from the rest of the year without selecting any day from January, divided by the total number of all possible sets of 60 days. Correct.
d) 1/12 - This is the probability of selecting any day from the year that is not from January. Incorrect.
Answer: c) (330 C 60) / (360 C 60)
3) There exist 3 different months such that exactly 20 days are selected from each one of these months.
a) (30 C 20)^3 / (360 C 60) - This is the probability of selecting exactly 20 days from one month cubed, divided by the total number of all possible sets of 60 days. Incorrect.
b) (90 C 60) / (360 C 60) - This is the probability of selecting any 60 days from three months, divided by the total number of all possible sets of 60 days. Incorrect.
c) ((12 C 3) * (30 C 20)^3) / ( 360 C 60) - This is the probability of selecting exactly 20 days from three different months (which can be chosen in 12 C 3 ways) cubed, divided by the total number of all possible sets of 60 days. Correct.
d) 1/4 - This is an arbitrary probability and does not consider the specific requirements given. Incorrect.
Answer: c) ((12 C 3) * (30 C 20)^3) / ( 360 C 60)