Problem 36 - Algebra 2 and Trigonometry

Question

Stan has letter tiles AMTE. a. How many different ways can Stan arrange all four tiles? b. How many different arrangements begin with a vowel?

Step-by-Step Solution

Answer

a) 24 ways. b) 12 ways.

Step 1: Identifying the Problem

We need to determine the number of permutations of a set of four distinct letters, and also figure out how many of these permutations start with a vowel. The tiles are A, M, T, and E.

Step 2: Calculating Total Arrangements

Stan has four unique tiles: A, M, T, and E. The total number of arrangements is the number of permutations of these four tiles. This can be calculated with the factorial function, \[ 4! = 4 \times 3 \times 2 \times 1 = 24. \] There are 24 possible ways to arrange all four tiles.

Step 3: Identifying Vowels

The vowels in Stan's tiles are A and E. We need to find how many arrangements begin with either of these vowels.

Step 4: Calculating Arrangements Starting with 'A'

If a tile arrangement starts with 'A', the remaining tiles are M, T, and E. The number of arrangements for these tiles is \[ 3! = 3 \times 2 \times 1 = 6. \] Hence, there are 6 different ways to arrange the tiles starting with 'A'.

Step 5: Calculating Arrangements Starting with 'E'

Similarly, if the arrangement starts with 'E', the remaining letters M, T, and A can be arranged in \[ 3! = 3 \times 2 \times 1 = 6 \] ways. Thus, there are 6 different arrangements starting with 'E'.

Step 6: Summing Arrangements Starting with Vowels

Add the number of arrangements starting with 'A' and 'E': \[ 6 + 6 = 12. \] Therefore, there are 12 different arrangements that begin with a vowel.

Key Concepts

Vowels

In the context of permutations involving letter tiles, vowels play a critical role, especially when you want specific arrangements that begin with one. In the English language, vowels are the sound-making letters such as A, E, I, O, and U. For the set of letter tiles AMTE, the vowels present are 'A' and 'E'.
When determining how many permutations of the tiles begin with a vowel, identifying the vowels in the set is the first step. By focusing on these vowels, we can easily calculate specific arrangements—those that specifically start with one of these letters. This makes vowels distinct as they set a particular group of starting positions for creating arrangements.

Factorial Function

The factorial function is a crucial mathematical operation often used in permutations and combinations. Denoted by an exclamation mark, the factorial of a number is the product of all positive integers less than or equal to that number. For example, \[ n! = n \times (n-1) \times \ldots \times 2 \times 1 \].
In problems involving permutations, like arranging letter tiles, the factorial function helps determine how many different ways a set of items can be ordered. For Stan’s tiles, the total number of different arrangements of his four tiles AMTE is given by the factorial of 4, \[ 4! = 24 \].
This indicates there are 24 unique ways to arrange the tiles. The factorial function simplifies calculation when dealing with multiple elements, streamlining the process of solving permutation problems.

Arrangements

When you think about arrangements in the context of letter tiles, you are essentially considering permutations, which are possibilities of arranging a set of items. Each unique order of the letters represents a different arrangement. This concept assumes that all elements are used and none are repeated in any single arrangement.
For Stan's four letter tiles, AMTE, an arrangement involves using each letter exactly once to form a sequence. Understanding arrangements is key to solving problems that ask for how many ways you can order or organize a set of items. By counting the total number of arrangements or focusing on specific conditions, such as arrangements that start with a vowel, we use permutations as a tool of calculation.

Letter Tiles

Letter tiles offer a tactile way to understand concepts like permutations and combinations. By physically moving and rearranging tiles, we can visually interpret the number of possible arrangements. In Stan's scenario, each letter tile—A, M, T, or E—can be thought of as a unique piece.
Considering letter tiles helps ground abstract mathematical ideas in a more hands-on format. This can be especially helpful in educational settings where visual and physical interaction aids in comprehension. Letter tiles allow students to experiment with different sequences, fostering a better grasp of how permutations work in practice. For problems involving permutations of letter tiles, like arranging them in specific orders, they make the problem both tangible and relatable.