Questions: How many different five-letter passwords can be formed from the letters A, B, C, D, E, F, G, and H if no repetition of letters is allowed? five-letter words can be formed.

Solution Steps

To determine how many different five-letter passwords can be formed from the given letters without repetition, we need to calculate the number of permutations of 5 letters chosen from a set of 8 distinct letters. This can be done using the permutation formula \( P(n, r) = \frac{n!}{(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

We have a total of \( n = 8 \) distinct letters: A, B, C, D, E, F, G, and H.

We need to form a password consisting of \( r = 5 \) letters.

To find the number of different five-letter passwords that can be formed without repetition, we use the permutation formula:

\[ P(n, r) = \frac{n!}{(n-r)!} \]

Substituting the values:

\[ P(8, 5) = \frac{8!}{(8-5)!} = \frac{8!}{3!} \]

Calculating this gives:

\[ P(8, 5) = \frac{8 \times 7 \times 6 \times 5 \times 4}{1} = 6720 \]

Final Answer