In how many distinct permutations can the letters of the word TRANSFORMER be arranged if the letters N and S always appear together as adjacent letters (in either order)?

Introduction / Context:
This question involves permutations of a word that has repeated letters and an additional restriction that two specified letters must always appear together. The word is TRANSFORMER and we are asked to count arrangements where N and S are adjacent in the sequence.

Given Data / Assumptions:

• Word: TRANSFORMER.
• Letters: T, R, A, N, S, F, O, R, M, E, R.
• The letter R appears three times; all other letters appear once.
• The letters N and S must always be together as NS or SN.

Concept / Approach:
We treat the pair N and S as a single block that can internally be arranged in 2 ways (NS or SN). Then we arrange this block together with the remaining letters, taking into account that R is repeated three times. This is a standard block and repetition problem in permutations.

Step-by-Step Solution:
Step 1: Remove N and S from the list and treat them as a block X. Step 2: The remaining letters are T, R, A, F, O, R, M, E, R and the block X. That gives a total of 10 objects. Step 3: Among these 10 objects, the letter R occurs three times, while all other objects are distinct. Step 4: Number of permutations of 10 objects with 3 identical R letters is 10! / 3!. Step 5: Inside the block X, the pair can be arranged as NS or SN, giving 2 internal arrangements. Step 6: Total arrangements = (10! / 3!) * 2. Step 7: Compute the value: 10! = 3628800 and 3! = 6, so 10! / 3! = 604800. Multiply by 2 to get 1209600.

Verification / Alternative check:
You can double check the count by confirming the multiset structure and ensuring that only the three R letters are considered identical. No further division is needed because all other letters and the NS block are distinct. The final multiplication by 2 accounts correctly for both NS and SN orderings.

Why Other Options Are Wrong:
112420, 85120 and 40320 are all significantly smaller than the correct figure and typically come from ignoring either the repeated R letters or the block constraint, or from miscounting the total number of objects to arrange.

Common Pitfalls:
Students may forget to divide by 3! for the three R letters or fail to multiply by 2 for the internal arrangement of N and S. Others may mistakenly treat N and S as always in the same internal order, which halves the correct count.

Final Answer:
The required number of permutations of TRANSFORMER with N and S always together is 1209600.