Does the order in which we choose letters matter here or can we use the combination rule?
Here, we have 6 different things, and we are taking 4, 5, or all 6 of these things and arranging them in as many ways as possible. Each ordering makes a different password, so the order in which we choose letters definitely matters here!
We are finding the possible permutations of six different letters in 4, 5, or 6 positions. So we can use the fundamental counting principle as in the text explanation. We could also think of this in terms of the permutations formula:
6P6 + 6P5 + 6P4 =
(6!/(6-6)!) + (6!/(6-5)!) + 6!/(6-4)!) =
6!/0! + 6!/1! + 6!/2! =
6! + 6! + 6!/2 = 1800
Now, if the order of letters didn't matter and we were just choosing groups of 4, 5, or 6 letters, we would have far fewer ways to do this. If the question said: "How many different groups of letters can be chose from the letters MAGOSH if a group must have at least 4 letters?"
Then our solution could use the combinations formula:
6C6 + 6C5 + 6C4 =
(6!/6!0!) + (6!/5!1!) + 6!/2!4!) =
1 + 6 + 15 = 22
If you are selecting a group of things from a larger group of things and the order in which you select those things does not matter, then you have a combination problem. In our original problem here, the order in which we choose letters matters, because MAGOSH is different code from AMGOHS etc, so we are looking at permutations, and not combinations.
Here are a couple blog posts you might find helpful: