Not an answer but a frequent silly mistake :
for conflict, you have 10 slots and 3 courses, place 2 courses in a single slot → conflict
pick 1 out of 3 (don’t care) and place it anywhere and for the rest 2 place it in the same slot
= [3, pick one out of 3] * [10, but don't care Course] * [10, place 2 in single one]
= 3* 10 * 10 = 300
now we use the inclusion-exclusion principle as all 3 in a single slot case, ie in any one of 10 slots,
= 300 - 10
= 290
so probability = 0.29
this Classic example of over-counting, since we counted (all 3 elements in the single slot) → 3 times, but we only removed it a single time,
[when we picked 3 courses say C1 → we get all 3 courses in a single slot, it also repeats for C2 and C3] , counting it “3 times“ so we have to remove it one more time (2 times in total), or we can completely remove it and count it as = 3 * (10 * 10 – 10 ) = 3*(90)
= 270 and now add all 3 in a single case
= 270 + 10 = 280
so, probability = 0.28
why mess up things, when we can use the Complement method, ie count negation(required) and remove it from total possibilities,
- total = 1000
- no conflict = place each one in different slot = 10 * 9 * 8 = 720
- required = 1000 – 720 = 280
- hence probability = 0.28 ( yes they asked probability, don’t answer it 280)
