Explanation:

The Indian Constitution does not contain any specific procedure for the selection and appointment of the Chief Minister. Article 164 only says that the Chief Minister shall be appointed by the Governor.
Statement 1 is correct: The Governor may have to exercise his individual judgement in the selection and appointment of the Chief Minister when the Chief Minister in office dies suddenly and there is no obvious successor. However, on the death of a Chief Minister,the ruling party usually elects a new leader and the Governor has no choice but to appoint him as the Chief Minister.
Statement 2 is correct: The Governor may first appoint him as the Chief Minister and then ask him to prove his majority in the Legislative Assembly within a reasonable period.
Statement 3 is correct: A person who is not a member of the State Legislature can be appointed as the Chief Minister for six months, within which time, he should be elected to the State Legislature, failing which he ceases to be the Chief Minister.
Statement 4 is correct: According to the Constitution, the Chief Minister may be a member of any of the two Houses of a State Legislature. Usually Chief Ministers have been selected from the Lower House (Legislative Assembly), but, on a number of occasions, a member of the Upper House (Legislative Council) has also been appointed as Chief Minister.