Question

A "word" is a string of one or more lowercase letters. How many words can be formed using all the letters of the word hyperbola? In how many words will \(h\) and \(y\) occur together? In how many will \(h\) and \(y\) not occur together?

Short answer

Total words: 362880; 'h' and 'y' together: 80640; 'h' and 'y' not together: 282240.

Step-by-step solution

Understanding the Problem

We need to calculate the total number of distinct words formed using all letters in 'hyperbola'. The word consists of 9 different letters: h, y, p, e, r, b, o, l, a.

Calculating Total Words

The total number of distinct permutations (words) that can be formed with 9 distinct letters is \[ 9! = 362880. \]

Finding Words with 'h' and 'y' Together

To find the number of words with 'h' and 'y' together, we treat 'hy' as a single block or unit. This means we now have 8 units (hy, p, e, r, b, o, l, a) to arrange.The number of permutations of these 8 units is \[ 8! = 40320. \]Within the 'hy' block, 'h' and 'y' can be arranged in 2 ways, as 'hy' or 'yh'.Therefore, the total number of arrangements where 'h' and 'y' are together is \[ 8! \times 2 = 40320 \times 2 = 80640. \]

Calculating Words with 'h' and 'y' Not Together

To find the number of words where 'h' and 'y' are not together, subtract the number of words where 'h' and 'y' occur together from the total number of words.So, we have \[ 9! - 8! \times 2 = 362880 - 80640 = 282240. \]

Key concepts

Combinatorics

Combinatorics is the branch of mathematics that deals with counting combinations and arrangements. It's a fundamental concept in various fields, especially mathematics, computer science, and statistics. In this problem, we used combinatorics to solve two main tasks: finding all possible 'words' using letters from a word "hyperbola" and determining specific scenarios where certain letters are grouped together or apart.

Combinatorics helps in organizing complex problems into manageable parts by understanding arrangements, permutations, and combinations. In our exercise, the term "permutations" is vital. We approach it from the perspective of permutations since the letters are distinct and we are arranging them in different ways. In this context, permutations involve different sequences of the entire set of letters. This is crucial since the order of letters creates different sequences, making each a unique 'word'.

Understanding how to rearrange or cluster certain elements, like treating 'h' and 'y' as a unit or block, also showcases combinatorics' flexible application. This allows us to alter the problem's constraints while maintaining a systematic counting approach.

Probability

Probability in this context dictates how likely it is to encounter a specific arrangement of the letters in 'hyperbola.' While we don't directly calculate probability values here, understanding basic probability principles is beneficial. Probability often goes hand-in-hand with combinatorics, as it helps us understand the frequency of a particular event over all possible outcomes.

When we find the number of arrangements where 'h' and 'y' are together or not, we're essentially determining the different "outcomes" of possible scenarios where these letters appear in formation. Understanding this helps us conceptualize scenarios and prepare for future probability-based questions.

In similar exercises, finding probability might involve calculating the ratio of favorable outcomes (words with 'h' and 'y' together) to the total number of outcomes (all possible permutations). This ratio gives us an intuitive sense of likelihood, providing deeper insights into various ways of arranging elements systematically.

Factorial

The factorial operation, denoted as \(!\), is a fundamental element in counting processes, especially in permutations and combinations. It's essential in our problem where we calculated the total number of permutations of the letters in 'hyperbola'.

Factorial is the product of all positive integers up to a certain number. For example, \(9!\) equals the product of numbers from 1 to 9. This plays a crucial role in understanding how many ways we can arrange these distinct letters.

Why factorial? Because each position in our word can be occupied by any remaining letter. The first position has 9 choices (since we have 9 distinct letters), then 8, and so on until the last position. Multiplying these choices together gives us the total permutations, hence the use of factorial. Understanding this concept helps students grasp complex arrangements concepts and see how small changes impact the entire counting process. The factorial helps manage large numbers in mathematics, bringing organization to combinatorial calculations.