Problem 10 - Algebra 2 and Trigonometry

Question

The letters of the word SEED are arranged at random. What is the probability that the arrangement begins and ends with E?

Step-by-Step Solution

Answer

The probability that the arrangement begins and ends with E is \( \frac{1}{6} \).

Step 1: Determine Total Arrangements

Calculate the total number of possible arrangements for the letters in the word SEED. There are 4 letters, but the letter 'E' is repeated twice, so we use the formula for permutations of multiset:\[ \frac{4!}{2!} = \frac{24}{2} = 12 \] Therefore, there are 12 possible arrangements for the letters.

Step 2: Find Desired Arrangements

Next, determine the number of arrangements where the word begins and ends with 'E'. We fix 'E' at the first and last positions, leaving the letters 'S' and 'D' to arrange in the middle.There are 2 positions left (second and third) and 2 letters ('S' and 'D'):\[ 2! = 2 \] This means there are 2 arrangements that fulfill this condition: ESD and EDS.

Step 3: Calculate the Probability

Now, calculate the probability using the formula:\[ \text{Probability} = \frac{\text{Number of Desired Outcomes}}{\text{Total Number of Outcomes}} \]Substitute the values:\[ \text{Probability} = \frac{2}{12} = \frac{1}{6} \] So, the probability of the arrangement beginning and ending with 'E' is \( \frac{1}{6} \).

Key Concepts

Arrangements

Arrangements refer to the different ways in which a set of items can be organized or ordered. It's an essential concept in probability and combinatorics, where the order of the elements is significant. When we talk about arrangements of letters or objects, we are trying to understand all possible sequences that can be created, given a set of components.

For instance, consider a set of letters. The arrangement of these letters is about figuring out how they can be placed in order to create unique sequences. If you have four distinct letters, the total number of ways to arrange them would be calculated by using the factorial notation, represented as 4!, which equals 24. However, when letters or items repeat, the total number of unique arrangements changes and needs special consideration, such as using the formula for permutations of a multiset.

To find such arrangements, it's crucial to account for these repetitions. This ensures that repeated elements do not inflate the possible outcomes when calculating probabilities.

Permutations of Multisets

Permutations of multisets involve arranging items where some elements are repeated. This differs from arranging unique items because repetitions affect the total number of distinct sequences.

In a multiset, the formula to calculate the number of permutations takes into account these repetitions. For instance, if you have a multiset with four items, where two items are identical, you calculate the permutations using the formula \( \frac{n!}{p_1!p_2!...p_k!} \). Here, \(n\) is the total number of items and \(p_1\.\.\.p_k\) are the factorials of the counts of the repeated elements.

Let's apply this to the word "SEED", where the letter 'E' repeats. Calculating permutations for this involves \( \frac{4!}{2!} \), leading to 12 distinct arrangements. This formula effectively reduces the total count by considering how many times identical arrangements would appear due to repeated elements.

Combinatorics

Combinatorics is a broad area of mathematics that involves counting, arrangement, and combination of objects. It helps in solving problems related to probability, statistics, and many more areas by providing tools to evaluate complex grouping and ordering scenarios.

Within combinatorics, key concepts include understanding arrangements (permutations) and selections (combinations). Arrangements focus on the order of elements, whereas combinations are concerned with the selection of a set of elements where the order doesn't matter. This distinction is crucial since permutations concern themselves with different sequences, while combinations focus merely on the presence of elements.

In this context, the exercise explores combinatorial principles by evaluating permutations and specializing in permutations of multisets. By calculating how many specific arrangements of the letters "SEED" are possible, and how many meet given conditions, students gain practical insights into making such combinatorial calculations accurately.