How can you tell whether a counting problem is a permutation or a combination problem? Give a real world example if possible

To distinguish between a permutation and a combination problem, you need to consider whether the order of the items you are selecting matters.

1. Permutations are used when the order of the items does matter. In a permutation, each arrangement of the items is unique. For example, if you are making a password out of letters and you are choosing 4 out of 26 letters, the order in which you arrange those letters is important, as 'ABCD' and 'DABC' would be considered different passwords.

2. Combinations are used when the order of the items does not matter. In a combination, you are concerned with the selection of items, but not the sequence in which they are arranged. For instance, if you are picking a committee of 4 people out of 20, it does not matter in what order you pick them; any selection of 4 people will be considered the same committee.

Real world example: Imagine you are at a smoothie shop, and you are allowed to choose 3 different fruits from a selection of 5 fruits to put in your smoothie. If the smoothie is blended up and the order of fruits doesn't affect the taste, then this is a combination problem because whether you choose Apple, Banana, Cherry or Cherry, Banana, Apple, the ingredients of the smoothie are the same.

On the other hand, if you are trying to unlock a 3-digit lock and you have 5 different numbers you can choose from for each digit, this is a permutation problem. The order of the numbers is crucial — 123 would unlock the lock, while 321 would not, even though both sequences use the same three numbers.