Questions: 3. How many friends must you have to guarantee at least five of them will have birthdays in the same month?

Solution Steps

To solve this problem, we can use the pigeonhole principle. There are 12 months in a year, and we want at least 5 friends to have birthdays in the same month. We can calculate the minimum number of friends needed by considering the worst-case scenario where the birthdays are distributed as evenly as possible across the months. In this case, each month would have 4 friends, and we would need one more friend to ensure that at least one month has 5 friends.

We need to determine the minimum number of friends required to ensure that at least five of them share a birthday in the same month. This can be approached using the pigeonhole principle.

Let \( n \) be the number of friends and \( m \) be the number of months. According to the pigeonhole principle, if we want at least one month to have at least \( k \) friends, we can distribute \( k-1 \) friends in each month without exceeding that number. Thus, we can have:

\[ \text{Maximum friends without 5 in the same month} = m \times (k - 1) \]

In our case, \( m = 12 \) (months) and \( k = 5 \) (friends), so:

\[ \text{Maximum friends} = 12 \times (5 - 1) = 12 \times 4 = 48 \]

To ensure that at least one month has 5 friends, we need to add one more friend to the maximum calculated above:

\[ \text{Minimum friends required} = 48 + 1 = 49 \]

Final Answer