Double counting

GRE Quant Problem Solving
1

Hi, I really struggle to see the difference between these three questions in terms of when we have to adjust for doble counting and when we don’t:

What is the number of distinct 3 3-digit numbers we can make with the numbers in set S={1,2,3,4,5}? Here we do not adjust for double counting according to GreMat’s response, but if we compute 5x5x5 are we not counting (if a=1 b=1 c=2) abc and bac and therefore should eliminate one?

In how many different ways can the letters in the word EMPIRE be ordered such that the first letter is a vowel? Here we do adjust for double counting of Es (predictable)

A box contains 1 red cube, 2 identical blue cubes and 2 identical green cubes. What is the number of distinct ways one can take all the cubes out of the box so that all blue cubes are removed before all green cubes are removed? And here we do adjust!!! While clearly the 2 blue boxes are two separate elements and not the same element.

2

We’re not double counting here (?)

Consider a smaller version of your problem at hand:

What is the number of distinct 2-digit numbers we can make with the numbers in Set S = {1,2}$

If you exhaustively write out all the possibilities, you have \{ (1,1), (1,2), (2,1), (2,2) \}. We have 2 \cdot 2 = 4 possibilities. Did we overcount?