The $120$ permutations of the word BORIS are arranged in alphabetical order, from BIORS to SROIB. What is the $60$th word in this list?
1. **Determine the number of permutations starting with each letter:**
   - The word "BORIS" has 5 distinct letters.
   - The number of permutations of the remaining 4 letters after choosing the first letter is \(4!\).
   - Therefore, there are \(4! = 24\) permutations for each choice of the first letter.

2. **Identify the range of permutations for each starting letter:**
   - Words starting with 'B': Positions 1 to 24.
   - Words starting with 'I': Positions 25 to 48.
   - Words starting with 'O': Positions 49 to 72.
   - Words starting with 'R': Positions 73 to 96.
   - Words starting with 'S': Positions 97 to 120.

3. **Determine the starting letter of the 60th word:**
   - Since the 60th word falls within the range of positions 49 to 72, it must start with 'O'.

4. **Determine the position within the 'O' starting words:**
   - The 60th word is the 12th word in the list of words starting with 'O' (since \(60 - 48 = 12\)).

5. **List the permutations starting with 'O':**
   - The remaining letters are B, I, R, S.
   - The number of permutations of these 4 letters is \(4! = 24\).
   - We need the 12th permutation of the letters B, I, R, S.

6. **Find the 12th permutation of B, I, R, S:**
   - The permutations of B, I, R, S in lexicographical order are:
     1. BIRS
     2. BISR
     3. BRIS
     4. BRSI
     5. BSIR
     6. BSR
     7. IBRS
     8. IBSR
     9. IRBS
     10. IRSB
     11. ISBR
     12. ISRB

7. **Combine the starting letter 'O' with the 12th permutation of B, I, R, S:**
   - The 12th permutation of B, I, R, S is ISRB.
   - Therefore, the 60th word is OISRB.

The final answer is $\boxed{OISRB}$