To solve this problem, we can use the formula for permutations with repetition, which is:
n^r
where n is the number of choices for each position and r is the number of positions.
For each of the following conditions, we will use this formula to determine the number of possible 5-letter words that can be formed using the given letters:
No letters can be repeated:
In this case, there are 7 choices for the first letter, 6 choices for the second letter (since one letter has already been used), 5 choices for the third letter, 4 choices for the fourth letter, and 3 choices for the fifth letter. Therefore, the total number of possible 5-letter words is:
7 x 6 x 5 x 4 x 3 = 2,520
Any letter can be repeated:
In this case, there are 7 choices for each of the 5 positions. Therefore, the total number of possible 5-letter words is:
7 x 7 x 7 x 7 x 7 = 16,807
Exactly one letter must be repeated:
There are two cases to consider: the repeated letter can be in the middle (ABCDD), or it can be at the end (ABCCD).
For the first case, there are 7 choices for the first letter, 6 choices for the second letter (since the first letter has already been used), 5 choices for the third letter (since it cannot be the same as the first two), and 1 choice for the repeated letter (since it must be the same as one of the first two letters). Therefore, the total number of possible words for this case is:
7 x 6 x 5 x 1 x 6 = 1,260