(The alphabet with 26 letters) How many words are there

1. of 5 letters?
2. of 5 different letters
3. of 5 letters but without the same letter twice in a row?
1. For each letter there are 26 possibilities. This gives $26 \times 26 \times 26 \times 26 \times 26=26^{5}$ possible words.
2. For the first letter there are 26 possibilities. For the second letter, it is forbidden to choose the first, which leaves 25 possibilities. For the third, only 24 remain, etc... There are then $26 \times 25 \times 24 \times 23 \times 22$ possibilities.
3. For the first letter there are 26 possibilities. For the second letter, it is forbidden to choose the same, which leaves 25 possibilities. The same applies to the third and subsequent letters; it is only necessary to choose a different letter from the previous one, so 25 possibilities. There are then $26 \times 25 \times 25 \times 25 \times 25$ possibilities.