Final answer:
The number of distinguishable arrangements for the letters in 'smith' is calculated using five factorial (5!), which equals 120. Option (b) 120 is the correct answer.
Step-by-step explanation:
Calculating the Number of Distinguishable Arrangements
To find the number of distinguishable arrangements for the letters in the word 'smith,' we use the factorial method. Since there are no repeating letters in 'smith,' we simply take the factorial of the number of letters. 'Smith' has 5 letters, hence we calculate 5! (five-factorial), which is 5×4×3×2×1.
If you systematically list all possible combinations, you'll notice that by fixing one letter and finding all the permutations of the remaining letters, you have a structured way of covering all arrangements without repetitions. The orderly manner in which you shuffle the letters contributes to understanding why the factorial of the number of letters gives the correct count of arrangements.
After calculating, 5! equals 120. Therefore, the number of distinguishable arrangements of the letters in 'smith' is 120. Among the provided options, (b) 120 is the correct answer.