The Birthday Problem (another probability question)

A room contains *n* randomly chosen people.

(a) Assume that a randomly chosen people is equally likely to have been born on any day of the week. The probability that the people in the room were all born on different days of the week is denoted by P.

(i) Find P in the case n=3.

(ii) Show that P=120/343 in the case n=4.

(b) Assume now that a randomly chosen people is equally likely to have been born in any month of the year. Find the smallest value of *n* such that the probability that the people in the room were all born on different months of the year is less than 0.5.

(c) Assume now that a randomly chosen people is equally likely to have been born on any of the 365 days in the year. It is given that, for the case n =21, the probability that the people in the room were all botn on different days of the year is 0.55631, correct to 5 places of decimals. Find the smallest value of n such that the probability that at least two of the people were born on the same day of the year exceeds 0.5.

I thought that for (a) part 1, it would be (3/7) x (2/6) x (1/5) but then I can't show for part 2 so I guess I'm wrong. And I have no clue on (b) and (c) at all! So thanks so much if you could help!