Problem 42 - Algebra and Trigonometry

Question

Explain why in any group of 677 people, at least two people must have the same pair of initials.

Step-by-Step Solution

Answer

Due to 676 initial combinations and 677 people, the pigeonhole principle guarantees duplicate initials.

Step 1: Understand the Problem Statement

We are asked to explain why in a group of 677 people, at least two people must share the same pair of initials. This involves understanding the concept of distribution and repetition.

Step 2: Determine Possible Initial Combinations

Each individual has a pair of initials, such as 'AB' for 'Alice Brown'. Assuming initials are composed of the 26 letters of the English alphabet, calculate how many unique initial combinations are possible.

Step 3: Calculate Total Initial Pairs

Since there are 26 letters and each person has a pair (first initial and last initial), we calculate the total possible pairs as follows:\[ 26 \times 26 = 676 \]

Step 4: Compare Total Initials to Number of People

We have 676 possible initial combinations and 677 people. By the pigeonhole principle, if the number of people exceeds the number of initial combinations, then at least two people must have the same initials.

Key Concepts

Initial Combinations

When we talk about initial combinations, we refer to the possible sets of initials that an individual can have. Initials are generally the first letters of a person's first and last names. Each letter can be one of the 26 letters of the English alphabet.
To find the total number of possible initial combinations, consider each initial separately. The first initial can be any letter from 'A' to 'Z', and similarly, the second initial can also be any letter from 'A' to 'Z'.
Therefore, the calculation for possible combinations is:

• For the first initial: 26 choices
• For the second initial: 26 choices
• Total combinations = 26 x 26 = 676

Thus, there are 676 unique combinations of initials possible. This number plays a crucial role when assessing how people's initials are distributed among a group.

Unique Initial Pairs

A unique initial pair is a specific combination of two initials, such as 'JD' for 'John Doe'. Unique pairs mean that no other combination has the same two letters in the same order.
In a situation involving initials, each of the 676 combinations can be considered unique. Unique initial pairs are significant when analyzing large groups of people to determine if any repetition occurs among the initials.

• With a total of 676 possible initial pairs, each one can represent a different combination.
• Each person's initials are initially assumed to belong to one of these pairs.

When everyone in a group of up to 676 people has a unique pair of initials, all combinations are perfectly distributed without any repetition. But, once the number of people exceeds 676, unique distribution is impossible.

Distribution and Repetition

The principle of distribution and repetition comes into play strongly when dealing with more people than available initial combinations. The Pigeonhole Principle is an essential concept to understand this distribution problem.
According to the Pigeonhole Principle, if you have more "pigeons" than "pigeonholes", at least one pigeonhole must contain multiple pigeons. In our context:

• Pigeons = People with initials
• Pigeonholes = Unique pairs of initials

When you have 677 people but only 676 possible initial pairs, it is guaranteed that at least two people will share the same pair of initials. This is because there's simply not enough unique initial pairs for everyone to have their own, given the constraints of the alphabetically-composed initial system. Thus, repetition of initials becomes inevitable.