How many friends must you have to guarantee that at least five of them will have birthdays in the same month?

Well, if there are 4 in each month, you have 48 friends.

The 49th will have to be in the same month as one of the others.

49

To determine the minimum number of friends needed to guarantee that at least five of them will have birthdays in the same month, you can use the Pigeonhole Principle. According to this principle, if you need to distribute m objects into n boxes, and m > n, then at least one box will contain more than one object.

In this case, the "objects" are the birthdays of your friends, and the "boxes" are the 12 months of the year. So, we need to find the minimum number of friends (objects) to guarantee that at least five of them will have birthdays in the same month (box).

To find this minimum number, we need to consider the worst-case scenario. The worst-case scenario would be if each of the 12 months is represented by only 4 friends (m = 4 * 12 = 48), and there are no more than 4 friends with birthdays in any given month.

Now, let's add one more friend to the group. Since there are only 12 months, and each month can have a maximum of 4 friends, the 49th friend must share a birthday with one of the previous friends in some month. Therefore, with 49 friends, we are guaranteed that at least five of them will have birthdays in the same month.

Therefore, the minimum number of friends needed to guarantee that at least five of them will have birthdays in the same month is 49.