In how many distinct permutations can the letters of the word RAILINGS be arranged if the letters R and S are always together as a single adjacent block (in either order)?

Introduction / Context:
This question tests permutations with a grouping restriction. We need to count how many different arrangements of the letters in the word RAILINGS are possible when the letters R and S must always appear next to each other, acting like a single block. This is a classic arrangement problem involving repeated letters and a special block condition.

Given Data / Assumptions:

• The word is RAILINGS.
• Letters: R, A, I, L, I, N, G, S.
• The letter I appears twice, all other letters are distinct.
• R and S must always be adjacent, either as RS or SR.

Concept / Approach:
We treat the pair (R, S) as a single block for counting, and remember that this block itself can be internally arranged in 2 ways (RS or SR). Then we arrange this block along with the remaining letters, taking care of the repeated letter I.

Step-by-Step Solution:
Step 1: Write the multiset of letters: {R, A, I, I, L, N, G, S}. Step 2: Combine R and S into a single block X. Inside the block, the order can be RS or SR. Step 3: Now the objects to arrange are: X, A, I, I, L, N, G. That is 7 items with I repeated twice. Step 4: Number of arrangements of these 7 items = 7! / 2! because of the two identical I letters. Step 5: For each such arrangement, the internal order of R and S within block X can be RS or SR, so multiply by 2. Step 6: Total arrangements = (7! / 2!) * 2 = 7! = 5040.

Verification / Alternative check:
Another way is to directly note that 2 in the numerator cancels the 2 in the denominator, leaving 7!. Since 7! = 5040, the count is consistent and there is no double counting or omission.

Why Other Options Are Wrong:
1260 and 2520 are smaller than 5040 and typically arise from dividing by extra factors that are not justified. 1080 is even smaller and does not match any meaningful factorial based count for this multiset and grouping condition.

Common Pitfalls:
Students often forget that R and S can be arranged in 2 ways inside their block or forget to divide by 2! for the repeated I. Some also mistakenly treat all letters as distinct, which gives 8! instead of the corrected grouped and repeated count.

Final Answer:
The number of permissible permutations of the letters of RAILINGS with R and S always together is 5040.