N2002 P2 Q30 (Either)

A room contains \(n\) randomly chosen people

1. Assume that a randomly chosen person 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 \(P\).
   1. Find \(P\) in the case \(n=3\).
   2. Show that \(P=\frac{120}{343}\) in the case \(n=4\).
2. Assume now that a randomly chosen person 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 in different months of the year is less than \(0.5\).[5]
3. Assume now that a randomly chosen person is equally likely to be born on any of the \(365\) days in the year. It is given that, for the case \(n=21\), the probability that the people were all born on different days of the year is \(0.55631\), correct to \(5\) places of decimals. Without the use of a graphic calculator, find the smallest value of \(n\) such that at least two of the people were born on the same day of the year exceeds \(\frac{1}{2}\).