Problem 30 - Algebra 2 and Trigonometry

Question

In \(23-30,\) find the number of different arrangements that are possible for the letters of each of the following words. MATHEMATICS

Step-by-Step Solution

Answer

There are 4,989,600 different arrangements of 'MATHEMATICS'.

Step 1: Count the Total Number of Letters

The word 'MATHEMATICS' has a total of 11 letters.

Step 2: Identify and Count Repeated Letters

In 'MATHEMATICS', the letters M, A, and T each appear twice. The remaining letters (H, E, I, C, S) each appear once.

Step 3: Calculate the Factorial of Total Letters

Calculate the factorial of the total number of letters: \[11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 39,916,800\]

Step 4: Calculate the Product of Factorials of Repeated Letters

For the repeated letters M, A, and T (each repeated twice), calculate the product of their factorials: \[2! \times 2! \times 2! = 2 \times 2 \times 2 = 8\]

Step 5: Calculate the Number of Different Arrangements

Use the formula for permutations of multiset to find the number of different arrangements:\[\frac{11!}{2! \times 2! \times 2!} = \frac{39,916,800}{8} = 4,989,600\]

Step 6: State the Final Result

The total number of different arrangements of the letters in the word 'MATHEMATICS' is 4,989,600.

Key Concepts

Factorial Calculation

Factorial calculation is an essential concept when determining permutations, especially when dealing with multisets. A factorial, denoted with an exclamation mark (!), represents the product of all positive integers up to a given number. For example, to compute the factorial of 11, we multiply all integers from 1 through 11 together:\[11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 39,916,800\]The factorial helps us understand the total number of ways in which a set of items can be arranged. This calculation is crucial in step 3 of the solution above, as it gives us the number of possible arrangements if all letters were unique. Understanding how to calculate factorials sets the groundwork for tackling more complex permutation problems, especially in combinatorial mathematics.

Repeated Letters

In many words, especially longer ones like 'MATHEMATICS', some letters can repeat. Repeated letters affect the overall number of unique arrangements since swapping these identical letters doesn’t create a new permutation. Therefore, we must adjust our calculations to account for these repetitions.For the word 'MATHEMATICS', the letters 'M', 'A', and 'T' are each repeated twice. To account for these repeated letters in our calculation, we find the factorial for each repeated letter count. This results in:

• \(2!\) because 'M' repeats twice
• \(2!\) because 'A' repeats twice
• \(2!\) because 'T' repeats twice

We then multiply these factorials together: \(2! \times 2! \times 2! = 8\). This product adjusts the total permutations to ensure that non-unique arrangements from these repeats aren't overcounted, ensuring the result reflects unique permutations only.

Arrangements of Letters

Arranging letters involves calculating the number of ways to organize them to form different words or sequences. With permutations of multisets, especially with repeated elements, there's a specific formula to apply. In the case of 'MATHEMATICS', we've identified the presence of repeated letters.To find the number of unique arrangements, we use the permutation formula for multisets:\[\frac{n!}{k_1! \times k_2! \times k_3! \ldots}\]Where \(n!\) is the factorial of the total number of letters, and each \(k_i!\) refers to the factorial of each group's repeated letters. For our example:\[\frac{11!}{2! \times 2! \times 2!} = \frac{39,916,800}{8} = 4,989,600\]This formula ensures only unique permutations are counted by dividing the total permutations by those that are repeated due to identical letters. The calculation yields 4,989,600 unique arrangements of 'MATHEMATICS', beautifully encapsulating how systematic accounting of each letter's repetition shapes the total possibilities.