Problem 32 - Algebra for College Students

Question

How many different 11-letter permutations can be formed from the 11 letters of the word MATHEMATICS? 4,989,600

Step-by-Step Solution

Answer

There are 4,989,600 different permutations.

Step 1: Identify Repetitions

First, we identify the different letters in the word 'MATHEMATICS' and how many times each is repeated. The word consists of the letters: M (2 times), A (2 times), T (2 times), H (1 time), E (1 time), I (1 time), C (1 time), S (1 time).

Step 2: Calculate Total Permutations Without Considering Repetitions

Calculate the number of permutations of 11 distinct letters, which is given by the factorial of 11, denoted as \( 11! \). This calculates the total permutations if there were no repeated letters.

Step 3: Adjust for Repeated Letters

To adjust for the fact there are repeated letters, divide by the factorial of the number of repetitions for each repeated letter. Therefore, the formula is: \[ \frac{11!}{2! \times 2! \times 2!} \], where \( 2! \) compensates for each set of repeated letters (M, A, T).

Step 4: Calculate the Expression

Calculate \( 11! = 39,916,800 \), then calculate \( 2! = 2 \). The expression becomes: \[ \frac{39,916,800}{2 \times 2 \times 2} \]. This simplifies to \( \frac{39,916,800}{8} = 4,989,600 \). Thus, there are 4,989,600 different permutations.

Key Concepts

Factorial Notation

Factorial notation is a mathematical concept used to express the product of all positive integers up to a certain number. It's denoted by an exclamation mark `!` following a number. For instance, the factorial of 5, noted as `5!`, is calculated as 5 × 4 × 3 × 2 × 1, which equals 120. This concept is central in counting and arranging objects, which is crucial in permutations and combinations.

The notation grows very quickly with large numbers. For example, `11!` is calculated as follows:

• 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.
• This results in a total of 39,916,800.

Factorials are essential in determining the number of possible arrangements, or permutations, of a set number of objects or letters when order is important.

Repetition in Permutations

Repetition in permutations is an important concept when dealing with arrangements of items that include duplicates. When items are repeated in a permutation, it reduces the number of unique arrangements. This is because repeated items cannot be distinguished from each other when swapped.

In the case of the word "MATHEMATICS," some letters repeat: M, A, and T, each appearing twice. To properly count the unique permutations of these letters, the total number of permutations, calculated using factorial notation, must be divided by the factorials of the counts of each repeated letter. This division adjusts for the over-counting of indistinguishable arrangements.

• For example, if a letter repeats twice, like M (2!), it contributes a factor of 2! (which is 2) in the denominator.
• This ensures calculations only account for those arrangements where the repeating letters are considered indistinguishable.

This adjustment helps us find the number of distinct permutations of the collection of letters.

Combinatorial Mathematics

Combinatorial mathematics is a branch of mathematics focused on counting, arranging, and optimizing configurations of objects. This field is widely applied in areas such as probability, statistics, computer science, and more. It helps in solving problems related to permutations and combinations, where order and repetition can significantly alter outcomes.

In our example of forming permutations from the word "MATHEMATICS," combinatorial principles guide us through understanding and solving the problem. The process involves determining the total possible arrangements (using factorial notation) and adjusting for repeats (using the concept of repetition in permutations) to find only the distinct permutations.

• Combinatorial mathematics provides tools and formulas that apply to various scenarios, making it invaluable for problem-solving in everyday contexts.
• It leverages fundamental principles, such as the multiplication and addition rules, to systematically approach complex arrangements and selections.

Thus, combinatorial mathematics is a key player in developing solutions across diverse fields where counting and decision-making are crucial.