P was born in July. Q’s birthday is before N. Q wasn’t born in July. Birthdays of both Q and N are in the same month. Therefore, both Q and N may have their birthdays in January, April or May. 
T’s birthday is immediately after N’s. The number of persons who have their birthdays between the birthdays of S and T is equal to the number of persons who have their birthdays between the birthdays of N and P. If N’s birthday is on 23 rd May or on 23 rd January, then as shown in figure, the condition of the equal no. of birthdays between S and T, and N and P won’t be met.(refer to the below table) 
The number of persons who have their birthdays between the birthdays of S and T is equal to the number of persons who have their birthdays between the birthdays of N and P. R does not have birthday in May. There are three birthdays between the birthdays of R and O. Now there are two possible dates for P’s birthday u.q. 14th July and 23rdJuly. First let us assume P's birthday on 14th July- 
But, the condition that there are three birthdays between the birthdays of R and O wouldn’t be met in the above arrangement. Therefore, assuming P’s birthday is on 23rd July and redrawing the table, we get- 
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