Generating 4-Letter Words from HISTORY: A Comprehensive Guide

The word 'HISTORY' comprises various combinations of consonants and vowels. In this article, we will explore how many 4-letter words can be generated from the letters of 'HISTORY,' with specific conditions on their beginning and ending letters. We will also discuss different methods to solve this problem, including the permutation and combination approach.

Understanding the Problem

The task is to find all possible 4-letter words that can be formed from the letters of 'HISTORY' under the following conditions:

The first letter must be a consonant. The last letter must be a consonant. The middle two letters can be either consonants or vowels.

Let's start by identifying the consonants and vowels in 'HISTORY.'

Consonants and Vowels in HISTORY

Consonants: H, S, T, R (4 consonants)

Vowels: I, O, Y (3 vowels)

Step-by-Step Calculation

We can break down the problem into steps, similar to the method provided in the given content:

Selecting the First and Last Letters

The first and last letters must be consonants. There are 4 choices for each (H, S, T, R).

Therefore, the number of ways to choose the first letter is 4, and the number of ways to choose the last letter is also 4 (considering repeats).

Selecting the Middle Two Letters

Each of the middle two letters can be any of the seven original letters (H, I, S, T, O, R, Y).

Thus, there are 7 possible choices for the third letter and 7 possible choices for the fourth letter.

Total Calculation

Let's calculate the total number of combinations:

Choices for 1st letter: 4
Choices for 2nd letter: 7
Choices for 3rd letter: 7
Choices for 4th letter: 4

Total combinations: 4 × 4 × 7 × 7 784

Therefore, the total number of 4-letter words that can be formed from 'HISTORY' under the given conditions is 784.

Alternative Methods to Solve the Problem

There are two alternative methods to solve this problem:

Permutation and Combination

As mentioned in the provided content, if repetition of letters is allowed and illegitimate words are considered, the total number of words can be calculated as follows:

Total number of words 7 x 7 x 7 x 7 2401 words

This method considers the letters of 'HISTORY' as independent entities, allowing any letter in any position.

Vocabulary Approach

Another approach, as suggested by Nikhil Lunia, includes finding all possible 4-letter words without repeating any letter. Here are some legitimate 4-letter words that can be formed:

Tyro Troy Trio Toys Tosh Tors This Rosy Rhos Riot Hots Host Invit Host His

Nikhil notes that there are 14 such words. He invites fellow Quorans to add more words to this list.

Conclusion

This guide covers different methods to generate 4-letter words from the letters of 'HISTORY,' each with specific conditions. Whether using the permutation and combination approach or the vocabulary method, the number of possible words varies greatly. Understanding the problem from multiple angles can provide a comprehensive view of the task.