Final answer:
The number of distinct ways the letters 'YYYSUY' can be arranged is 120. This is calculated using combinatorics by dividing 6! (the total arrangements for all unique letters) by 3! (to account for the three repeated 'Y' letters).
Step-by-step explanation:
To determine how many distinct ways the letters 'YYYSUY' can be arranged, we use combinatorics. We have a total of 6 letters where 'Y' is repeated 3 times and the other letters ('S' and 'U') are not repeated. The formula for finding the number of arrangements of n items where there are duplicates is n! divided by the factorial of each duplicate count. In this case, we have 6 letters, so 6! is the total number of arrangements if all letters were unique. Since 'Y' is repeated 3 times, we divide by 3! to account for the repetition.
6! is 6×5×4×3×2×1, which equals 720, and 3! is 3×2×1, which equals 6. Therefore, the total number of distinct arrangements of 'YYYSUY' is 720 divided by 6, resulting in 120 distinct arrangements.