How many 3 letters-long codes can be made by 5 different letters? You have five letters: C, H, E, S, T
How many different codes, consisting of three letters, can be made from the above letters?
I'd say ${5}\choose{3}$ is the correct answer, since the order of the letters doesn't matter. Is this (that the order doesn't matter) why $\frac{5!}{3!}$ isn't the correct answer?
 A: Regardless of interpretation, you can arrive at the answer using the Multiplication Principle.

If a every outcome can be uniquely described via a sequence of answers to questions with number of choices $a_1, a_2, a_3,\dots$ respectively where regardless of which choice is made the number of available choices doesn't change, then the total number of outcomes will be $a_1\cdot a_2\cdot a_3\cdots$


If letters are allowed to be repeated and order does matter:
Pick the first letter (5 choices), pick the second letter (5 choices), pick the third letter (5 choices), for a total of $5^3$ different codes.

If letters are not allowed to be repeated and order does matter:
Pick the first letter (5 choices), pick the second letter (4 remaining choices), pick the third letter (3 remaining choices), for a total of $5\cdot 4\cdot 3 = \frac{5!}{2!}$ different codes.

If letters are allowed to be repeated and order doesn't matter:
Apply stars-and-bars to get that there will be $\binom{3+5-1}{5-1}$ different codes.

If letters are not allowed to be repeated and order doesn't matter:
Standard combinations problem: there are $\binom{5}{3}$ such choices.  (Provable via a multiplication principle + division by amount of symmetry argument).
A: Letters can be repeated. The first letter can be chosen from 5 different letters, so can the second and third letter. Thus the answer is $5^{3} = 125$ different codes.
Thank you @Yinon Eliraz!
