In how many different ways can the letters of the word 'RITUAL' be arranged, assuming that all letters are used each time?

Introduction / Context:
This is a basic permutation problem involving all the letters of a word, with no additional restrictions. The word RITUAL consists of six distinct letters, and we want to count all possible different arrangements (permutations) of these letters. This type of question reinforces the simple factorial-based formula for permutations of distinct objects and forms the foundation for more complex arrangement problems.

Given Data / Assumptions:
- Word: RITUAL.

- Total letters: 6 (R, I, T, U, A, L), all distinct.

- Each arrangement (word) uses all 6 letters exactly once.

- Order of letters matters: different orders represent different arrangements.

- There are no repeated letters and no positional restrictions.

Concept / Approach:
When arranging n distinct items in a sequence, the number of different permutations is n! (n factorial). This comes from the fact that there are n choices for the first position, (n - 1) choices for the second, and so on, down to 1 choice for the last position. Multiplying these choices gives n! possible arrangements. Here, n = 6, so we simply compute 6!.

Step-by-Step Solution:
Step 1: Count the total number of letters in the word RITUAL.Step 2: There are 6 distinct letters.Step 3: The number of permutations of 6 distinct letters is 6!.Step 4: Compute 6! = 6 * 5 * 4 * 3 * 2 * 1.Step 5: Multiply stepwise: 6 * 5 = 30; 30 * 4 = 120; 120 * 3 = 360; 360 * 2 = 720; 720 * 1 = 720.Step 6: Therefore, there are 720 different arrangements.

Verification / Alternative check:
You can think of it position-wise: for the first position, any of the 6 letters can be used. After choosing the first letter, 5 letters remain for the second position, 4 for the third, and so on, down to 1 for the last position. Thus, the total number of sequences is 6 * 5 * 4 * 3 * 2 * 1, which again equals 720. This confirms that the factorial formula matches the intuitive counting process.

Why Other Options Are Wrong:
- 5040 equals 7!, which would be the number of permutations of 7 distinct items; this is not applicable here because RITUAL has only 6 letters.

- 360 is 6! / 2 and might arise if someone mistakenly divides by 2, perhaps thinking of symmetry or repetition where none exists.

- 180 is even smaller and could result from prematurely stopping the multiplication or accidentally dividing by additional factors.

Common Pitfalls:
Some learners miscount the letters or incorrectly assume there is a repeated letter in the word, which would require dividing by factorials of repeat counts. Others simply confuse 5! and 6!, underestimating the total. Being careful to identify the exact number of distinct symbols and using the correct factorial is crucial.

Final Answer:
The letters of the word RITUAL can be arranged in 720 different ways.