The word "sleepless" has 8 letters. To find the number of distinguishable permutations, we can use the formula for permutations of a set with no repeated elements, which is n!, where n is the number of elements.
Therefore, the number of permutations for the word "sleepless" can be calculated as 8!, which is equal to 40,320. This means that there are 40,320 different ways we can arrange the letters in the word "sleepless" while keeping all the letters distinct.
Note that if the word had repeated letters, we would have to divide the result by the factorials of the number of times each letter was repeated.