Find the minimum number of students needed to guarantee that 3 of them:

5

CO2

(a) have last names which begin with the same first letter;

(b) were born on the same day of a month (with 31 days).

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? 

Here are 

Find the minimum number of students needed to guarantee that 3 of them: (a) have last names which begin with the same first letter; (b) were born on the same day of a month (with 31 days).

Discover how to calculate the minimum number of students required to ensure that 3 of them have last names beginning with the same letter or were born on the same day in a 31-day month. Using the Pigeonhole Principle, this problem explains how 53 students are needed for part (a) and 63 for part (b). Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation