Translate birthday counts and find a strict minimum sample size

It may be assumed that the dates of birth in a large population are distributed throughout the year so that the probability of a randomly chosen person’s date of birth being in any particular month may be taken as $\dfrac1{12}$.

(i) Find the probability that, of 6 people chosen at random, exactly 2 will have birthdays in January.3 marks

(ii) Find the probability that, of 8 people chosen at random, at least one will have a birthday in January.3 marks

(iii) $N$ people are chosen at random. Find the least value of $N$ such that the probability that at least one will have a birthday in January exceeds 0.9.3 marks

(iv) Find the probability that, of 100 people chosen at random, at least 40 will have birthdays in May, June, July or Agusut.4 marks