you have 47 people.
you want to know at least how many were born in the same month.
i would think that at least 4 people had to be born in the same month.
here's why.
12 months in a year.
if every 12 people were born in a different month, then out of 47 people, at least 4 people were born in the same month.
the arrangement of all 47 people would look like this:
first 12 people 1 2 3 4 5 6 7 8 9 10 11 12
second 12 people 1 2 3 4 5 6 7 8 9 10 11 12
third 12 people 1 2 3 4 5 6 7 8 9 10 11 12
remaining 11 people 1 2 3 4 5 6 7 8 9 10 11
you could have a minimum of 3 people that were born in the same month, but that doesn't answer the question.
the question is at least how many people were born in the same month.
that would have to be 4 because there is at least one case where 4 people were born in the same month.
assume there were 37 people, than what is the least number of people that were born in the same month?
here's the complete set of 37 people.
first 12 people 1 2 3 4 5 6 7 8 9 10 11 12
second 12 people 1 2 3 4 5 6 7 8 9 10 11 12
third 12 people 1 2 3 4 5 6 7 8 9 10 11 12
remaining 11 people 1
it's a little easier to see here. there are a lot of people where 3 were born in the same month, but there is at least 1 case where 4 people were born in the same month, so the least number of people that were born in the same month still has to be 4.
that's my thinking, anyway.
your second problem:
(b) Assuming that no one is born on Feb. 29 (leap day), how many people should be selected to guarantee that at least 6 were born on the same day, not considering the year?
there are 365 days in the year.
assuming everybody was born on a different day, then you could have 365 people where nobody was born on the same day.
the 366 person, however, would have to have been born on the same day as somebody else in the group.
therefore, the minimum number of people in the group would have to be 366 to guarantee that at least 2 people were born on the same day.
but you wanted to guarantee that at least 6 people were born on the same day.
assuming you had 5 * 365 people together, with every 5 of them being born on the same day.
you would have a total of 365 * 5 = 1825 people, with no more than 5 people being born on the same day.
add 1 more person and you will be guaranteed that at least 6 people were born on the same day.
that comes out to be 1826 people.
1825 / 5 = 365 sets of people born on the same day, with each set containing no more than 5 people.
set 1 is the number of people born on day 1.
set 2 is the number of people born on day 2.
etc.
that next person had to be added to one of the sets of 5 to make it a set of 6.