How many of the following statements have to be true i No year can have 5 Sundays in the month of Ma...........

Quantitative Aptitude ( CCS) Set theory

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How many of the following statements have to be true?

i. No year can have 5 Sundays in the month of May and 5 Thursdays in the month of June.

ii. If Feb 14^th of a certain year is a Friday, May 14^th of the same year cannot be a Thursday

iii. If a year has 53 Sundays, it can have 5 Mondays in the month of May.

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Explanation:

Statement (I): No year can have 5 Sundays in the month of May and 5 Thursdays in the month of June.
A year has 5 Sundays in the month of May => it can have 5 each of Sundays, Mondays and Tuesdays, or 5 each of Saturdays, Sundays and Mondays, or 5 each of Fridays, Saturdays and Sundays. Or, the last day of the Month can be Sunday, Monday or Tuesday.
Or, the 1^st of June could be Monday, Tuesday or Wednesday. If the first of June were a Wednesday, June would have 5 Wednesdays and 5 Thursdays. So, statement I need not be true.
Statement (II): If Feb 14^th of a certain year is a Friday, May 14^th of the same year cannot be a Thursday
From Feb 14 to Mar 14, there are 28 or 29 days, 0 or 1 odd day
Mar 14 to Apr 14, there are 31 days, or 3 odd days
Apr 14 to May 14 there are 30 days or 2 odd days
So, Feb 14 to May 14, there are either 5 or 6 odd days
So, if Feb 14 is Friday, May 14 can be either Thursday or Wednesday. So, statement II need not be true.
Statement (III): If a year has 53 Sundays, it can have 5 Mondays in the month of May
Year has 53 Sundays => It is either a non-leap year that starts on Sunday, or leap year that starts on Sunday or Saturday.
Non-leap year starting on Sunday: Jan 1^st = Sunday, Jan 29^th = Sunday. Feb 5^th is Sunday. Mar 5^th is Sunday, Mar 26^th is Sunday. Apr 2^nd is Sunday. Apr 30^th is Sunday, May 1st is Monday. May will have 5 Mondays. So, statement III can be true.