Number of Ways to Arrange the Letters in the Word "dogmatic"

To find the number of ways to arrange the letters in the word "dogmatic," we can use the concept of permutations. A permutation is an arrangement of objects in a particular order.

Step 1: Counting the number of letters in the word

The word "dogmatic" has 8 letters in total.

Step 2: Identifying the repeating letters

In the word "dogmatic," the letters 'd' and 'o' appear twice.

Step 3: Calculating the number of ways to arrange the letters

To calculate the number of ways to arrange the letters, we can use the formula for permutations with repetition. The formula is:

P(n; n1, n2, ..., nk) = n! / (n1! * n2! * ... * nk!)

Where:
- n is the total number of objects (in this case, the total number of letters)
- n1, n2, ..., nk are the number of repetitions of each object (in this case, the number of repetitions of each letter)

In our case, n = 8, n1 = 2 (for 'd'), and n2 = 2 (for 'o').

Using the formula, we can calculate the number of ways to arrange the letters:

P(8; 2, 2) = 8! / (2! * 2!) = (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (2 * 1 * 2 * 1) = 20160 / 4 = 5040

Therefore, there are 5040 different ways to arrange the letters in the word "dogmatic."

Summary

To summarize, the number of ways to arrange the letters in the word "dogmatic" is 5040. This is calculated using the formula for permutations with repetition, which takes into account the number of repetitions of each letter in the word.