The word can only be arranged 1 way DOGMATIC. The letters can be arranged 8! ways = 40,320 ways

Number of Ways to Arrange the Letters in the Word 'DOGMATIC'

Introduction:
The word 'DOGMATIC' consists of 8 letters. We need to determine the number of ways these letters can be arranged. In order to do that, we can use the concept of permutations.

Permutations:
A permutation is an arrangement of objects in a specific order. In this case, we want to find the number of permutations of the letters in the word 'DOGMATIC'.

Calculating the Number of Permutations:
To calculate the number of permutations, we need to consider the following factors:

1. Total number of letters:
The word 'DOGMATIC' has a total of 8 letters.

2. Repeated letters:
Among the letters in 'DOGMATIC', we have two occurrences of the letter 'D' and two occurrences of the letter 'M'. When calculating permutations, we need to take into account the repeated letters.

Formula:
The formula to calculate the number of permutations with repeated letters is:

P = n! / (r1! * r2! * ... * rk!)

Where:
- P is the number of permutations
- n is the total number of objects
- r1, r2, ..., rk are the frequencies of the repeated objects

Calculations:
Using the formula, we can calculate the number of permutations for the word 'DOGMATIC':

P = 8! / (2! * 2!)

Simplifying this expression:

P = (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((2 * 1) * (2 * 1))

P = 40,320 / 4

P = 10,080

Conclusion:
There are 10,080 ways to arrange the letters in the word 'DOGMATIC' while considering the repeated letters.

8! ways that is 40320