A friend has a thing in a video game where she was trying to get items from a limited pool of 20 items that couldn't have any repeat and it took her 20 tries to get the item she wanted. People were saying there is a 1 in 20! chance of this happening, while I think there is just a 1 in 20 chance of this happening because while it is true that this exact permeation of events happening is 1 in 20! there are also 19! outcomes that have the same property so the chance of this happening.
My intuition was that if you had a deck of 19 black cards and 1 red card if the only information you care about is where the red card is then there is only 20 possible outcomes so the question we are asking is what is the chance that the red card is on the bottom of the deck and that is 1 in 20.