In how many different ways can all the letters of the word ABYSMAL be arranged, given that the letter A appears twice and all other letters are distinct?

Introduction / Context:
This question asks for the number of permutations of the letters of the word ABYSMAL. Here, one letter is repeated, which means we must adjust the basic factorial formula for permutations of distinct items to account for indistinguishable letters.

Given Data / Assumptions:

• The word is ABYSMAL.
• Letters are A, B, Y, S, M, A, L.
• The letter A appears twice; all other letters (B, Y, S, M, L) appear exactly once.
• We must use all letters in each arrangement.
• Arrangements that differ only by swapping the two As are not considered different.

Concept / Approach:
If there are n total letters, with one letter repeated r times and the rest distinct, then the number of distinct permutations is n! / r!. This divides out the overcount due to the indistinguishability of the repeated letters. Here we have 7 letters in total with A repeated 2 times, so we use 7! / 2!.

Step-by-Step Solution:
Step 1: Count total letters in ABYSMAL. There are 7 letters.Step 2: Count how many times each letter appears. A appears 2 times; B, Y, S, M, L appear 1 time each.Step 3: Use the formula for permutations with one repeated letter: total permutations = 7! / 2!.Step 4: Compute 7! = 5040.Step 5: Compute 2! = 2.Step 6: Divide: 5040 / 2 = 2520.

Verification / Alternative check:

Why Other Options Are Wrong:

• 5040: This would be 7! and treats both As as distinct, overcounting arrangements where As are swapped.
• 3650 and 4150: These values do not correspond to any simple factorial ratio and likely arise from arithmetic guesses rather than correct combinatorial reasoning.

Common Pitfalls:
A common mistake is to forget that the two As are identical and simply use 7!, which is wrong. Some students also incorrectly divide by more than 2!, imagining extra repetitions that do not exist. Carefully counting how many times each letter appears and then applying the n! divided by product of factorials of repeated counts is essential.

Final Answer:
The total number of different arrangements of the letters of ABYSMAL is 2520.