In how many different ways can all the letters of the word POVERTY be arranged to form distinct letter sequences?

Introduction / Context:
This question is a straightforward permutation problem where all letters in a word are distinct. We are asked to count the number of different sequences that can be formed using all letters of the word POVERTY, without any additional conditions.

Given Data / Assumptions:

• The word is POVERTY.
• Letters in POVERTY are P, O, V, E, R, T, Y.
• No letter is repeated; all 7 letters are distinct.
• We must use all letters in each arrangement.
• Different orders of the same letters count as different arrangements.

Concept / Approach:
When all letters are distinct and we use all of them, the number of arrangements equals the factorial of the number of letters. Since there are 7 distinct letters, the count of permutations is 7!.

Step-by-Step Solution:
Step 1: Count the number of distinct letters in POVERTY. There are 7 distinct letters.Step 2: To form an arrangement using all letters, we are essentially permuting 7 distinct items.Step 3: The number of permutations of 7 distinct items is 7!.Step 4: Compute 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1.Step 5: Calculate step by step: 7 * 6 = 42, 42 * 5 = 210, 210 * 4 = 840, 840 * 3 = 2520, 2520 * 2 = 5040.Step 6: Therefore, there are 5040 distinct arrangements.

Verification / Alternative check:

Why Other Options Are Wrong:

• 2520: This equals 7! / 2 and would be correct only if one letter were repeated twice, which is not the case here.
• 1260: This is 7! / 4 and does not correspond to any standard repetition structure in this word.
• None: The correct count is already represented by option 5040, so None is incorrect.

Common Pitfalls:
Students sometimes mistakenly think some letters repeat, for example confusing POVERTY with words that have double letters, and use formulas for permutations with repetitions. Another pitfall is arithmetic error when computing 7!. Writing out the factorial multiplication step by step and double checking helps avoid such mistakes.

Final Answer:
The number of different arrangements of the letters of POVERTY is 5040.