This is question in relation to the birthday paradox we discussed earlier.

A room contains $n$ randomly chosen people.

1. 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 $P$.
1. Find $P$ when $n=3$.
2. Show that $P=\frac{120}{343}$ when $n=4$
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 born in different months of a year is less than $\frac {1}{2}$.

Ans: $\frac{30}{49}; 5$