Problem 30 - Algebra and Trigonometry

Question

Making Words \(A\) toddler has wooden blocks showing the letters \(C, E, F, H, N,\) and \(R\) . Find the probability that the child arranges the letters in the indicated order. (a) In the order FRENCH (b) In alphabetical order

Step-by-Step Solution

Answer

Both probabilities are \(\frac{1}{720}\).

Step 1: Understand the Total Arrangements

To solve the problem, we first need to calculate the total number of ways to arrange six letters. This can be done using the factorial function: there are 6 different letters and they can be arranged in \(6!\) different ways. Calculate \(6!\).

Step 2: Calculate the Probability for Specific Arrangement (FRENCH)

The word 'FRENCH' is one specific arrangement among all possible arrangements. Hence, the probability of getting 'FRENCH' is \(\frac{1}{6!}\). Complete the division to find the probability.

Step 3: Calculate the Probability for Alphabetical Order

For a group of letters arranged in alphabetical order, each letter must strictly follow its alphabetical successor. With unique letters, there is exactly one way to arrange them alphabetically. Thus, the probability is \(\frac{1}{6!}\).

Step 4: Simplify Computations

Calculate the value of \(6! = 720\). Use this value to find both probabilities: \(\frac{1}{720}\) for both scenarios.

Key Concepts

Factorial Notation

Factorial notation is a mathematical concept used to denote the product of all positive integers up to a specified number. It is represented with an exclamation mark (e.g., 6!). In the context of counting, factorial notation helps determine how many ways you can arrange a set of different objects.
For example, if you have 6 letters like in our exercise, you compute 6! which equals 6 × 5 × 4 × 3 × 2 × 1. This translates to 720 different possible arrangements for those 6 letters.
Using factorials simplifies calculations in problems involving permutations, as it provides a systematic way to count arrangements without manually listing them all.

Permutations

Permutations refer to the arrangement of objects in a specific sequence or order. When dealing with permutations, the order of arrangement is crucial. With six distinct letters like our scenario, each different sequence of these letters represents a unique permutation.
To calculate permutations, we often use factorial notation, as it accounts for all possible sequences. When the order doesn't matter, combinations are considered, but in permutations, every arrangement is unique.
In our problem, each of the 720 arrangements of the letters C, E, F, H, N, and R is a different permutation. Specific permutations, like forming the word "FRENCH," can be seen as just one possible arrangement among the many.

Alphabetical Order

Alphabetical order involves arranging words or letters following the sequence of a standard order set by the alphabet. Each letter or set of letters follows or precedes another based on this order.
In the exercise, sorting the letters C, E, F, H, N, and R alphabetically means placing them from A to Z as it would appear in the dictionary. For unique letters, there is exactly one permutation where they fit perfectly into alphabetical order. That’s a key point in probability: only one out of all possible permutations will be the alphabetic set.
This is why arranging letters in alphabetical order results in a very specific arrangement, making the probability calculation relatively straightforward.

Probability Calculation

Probability is a measure of how likely an event is to occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
For example, determining the probability of the toddler arranging letters into the word 'FRENCH' involves identifying how many specific ways the word can be formed—just one—out of the 720 possible arrangements. This results in a probability of \( \frac{1}{720} \).
Similarly, the probability of the letters being arranged in alphabetical order is also \( \frac{1}{720} \), as there is only one specific sequence that fits this criterion. Such probability calculations play a critical role in understanding chance and randomness in different scenarios.