Something is very, very, very wrong here. The probability of the second person have a birthday on a day different than the first is 364/365 = 1 - 1/365. That lecturer needs to learn just a little bit to use intuition.
The lecturer suggests the probability that three random people have their birthdays on different days is (1/365) * (1/364) * (1/363). That's one in 48 million, much less likely than winning the lottery. I'll bet him $1,000, even money, that three random people have different birthdays, until he runs out of money.
It seems he is calculating the following probability: I guess the birthday of the first person. 1/365 that I get it right. Then I ask the next person if they have the same birthday until I find someone with a different birthday than the first person, and guess that second person's birthday. 1/364 that I get it right again. I find a third person with a birthday different from those two and guess their birthday. 1/363 that I get it right and so on.
(The quoted $Q_n$ seems to be the correct answer).