Let's solve the two parts of this problem step by step:

Part (a): Minimum number of students for 3 with last names beginning with the same letter

There are 26 letters in the English alphabet. We need to guarantee that at least 3 students have last names starting with the same letter. This is a classic application of the Pigeonhole Principle.

The Pigeonhole Principle states that if $n$ items are distributed among $k$ containers, then at least one container must contain more than one item if $n > k$.

For this case:

• There are 26 possible starting letters (containers).
• We want at least 3 students to share the same letter.

If we assume the worst-case scenario, we can place 2 students per letter without reaching 3 students with the same letter. This would give us: $26 \times 2 = 52 \text{ students}$ In this case, each letter is shared by exactly 2 students, so we still don't have 3 students sharing the same letter.

To guarantee that at least one letter is shared by 3 students, we need to add one more student: $52 + 1 = 53 \text{ students}$

Thus, 53 students are required to ensure that at least 3 have last names beginning with the same letter.

Part (b): Minimum number of students for 3 born on the same day of a month with 31 days

This is another application of the Pigeonhole Principle, but now the "containers" are the 31 possible days in a month.

We want to guarantee that at least 3 students were born on the same day. Again, the worst-case scenario would be placing 2 students per day across all 31 days: $31 \times 2 = 62 \text{ students}$ In this case, each day is shared by exactly 2 students, and no day has 3 students.

To guarantee that at least one day has 3 students, we need to add one more student: $62 + 1 = 63 \text{ students}$

Thus, 63 students are required to ensure that at least 3 were born on the same day of a month with 31 days.

Summary:

• 53 students are needed to guarantee that at least 3 have last names starting with the same letter.
• 63 students are needed to guarantee that at least 3 were born on the same day of a month with 31 days.

Would you like a more detailed explanation or have any further questions?

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