Problem 40 - Algebra and Trigonometry

Question

A toddler has eight wooden blocks showing the letters \(A, E, I, G, L, N, T,\) and \(R .\) What is the probability that the child will arrange the letters to spell one of the words TRIANGLE or INTEGRAL?

Step-by-Step Solution

Answer

The probability is \(\frac{1}{20160}.\)

Step 1: Identify Total Arrangements

To find the probability, we first need to determine the total number of ways the toddler can arrange the eight blocks. Because there are eight distinct letters, the total number of arrangements is the factorial of 8, denoted as 8!. Thus, the total possible arrangements is calculated as: \[8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320.\]

Step 2: Identify Desired Arrangements

Next, we need to count the number of arrangements that form the words TRIANGLE and INTEGRAL. Both words consist of the same eight letters and are complete anagrams of each other. Therefore, only two specific arrangements spell these words.

Step 3: Calculate the Probability

Now, calculate the probability that one of the two specific arrangements will occur out of all possible arrangements. The probability is given by the ratio of the number of successful arrangements to the total number of arrangements:\[\text{Probability} = \frac{2}{40320}.\]

Step 4: Simplify the Probability

The probability simplifies to: \[\frac{2}{40320} = \frac{1}{20160}.\]

Key Concepts

Factorial Calculation

The concept of factorial calculation is fundamental when determining possible arrangements of objects. When we talk about the factorial of a number, denoted as \( n! \), it means we are multiplying all the positive whole numbers up to \( n \). For example, with 8 blocks, like in our problem, the total arrangements will be the factorial of 8, represented as \( 8! \).Here’s how a factorial calculation works step-by-step:

• Start with the number itself (here, 8).
• Multiply it by each successive smaller number (7, 6, 5, and so on), until you reach 1.
• So, \( 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320 \).

This tells us that there are 40,320 different ways to arrange 8 distinct letters. Factorial calculations increase rapidly, highlighting how quickly possible arrangements multiply as objects increase, which is vital in probability and permutations calculations.

Anagram Probability

Anagram probability involves calculating the likelihood of a specific arrangement of letters forming a word from all possible letter arrangements. When dealing with letters like those in our exercise, forming the words TRIANGLE or INTEGRAL, we start by acknowledging they are anagrams.

• An anagram is a rearrangement of letters to form different words or phrases.
• The letters must exactly match those of a pre-defined word for the anagram to be successful.
• Here, each possible arrangement is equally likely if chosen randomly.

In the exercise, we are interested in the probability of forming either TRIANGLE or INTEGRAL. Since these two words are complete anagrams of each other, there are exactly 2 successful arrangements out of the 40,320 total possibilities. Knowing this helps us understand the probability associated with randomly reaching a specific order that forms a legitimate word.

Permutations and Combinations

Understanding permutations and combinations allows us to solve complex problems involving arrangements and selections of objects. Permutations are all about the arrangement of objects, where the order matters. In our exercise, we have permutations because we are shuffling 8 letters. Here’s how permutations are relevant:

• The main concern is the total number of ways to arrange all letters fully, using all slots, where each slot is filled once.
• Since each block is unique, every distinct arrangement, or permutation, is counted without repetition.
• Combinations come into play when we are not concerned with order but only with selecting groups; however, in this case, permutations are what dictate the complexity.

In permutations, every unique arrangement is significant, demonstrating how even slight changes to the order result in a fundamentally different array. This core understanding forms the foundation of probability calculations involving any permutation scenario, like arranging distinct blocks to form anagrams of actual words.