The question in the image is about constructing passwords using only lowercase letters of the English alphabet. Here's the detailed breakdown of the problem:

Question Details:

1. Condition: The password must be exactly four letters long.
2. Restrictions: Each letter can be used only once in the password (no repetitions are allowed).

Solution:

Step 1: Understand the total possibilities for each position.

• The English alphabet has 26 letters.
• For the first letter, there are 26 possible choices.
• For the second letter, since the first letter is already used, there are 25 remaining choices.
• For the third letter, there are 24 remaining choices.
• For the fourth letter, there are 23 remaining choices.

Step 2: Multiply the choices to calculate the total number of unique passwords.

The total number of passwords is given by: $26 \times 25 \times 24 \times 23$

Step 3: Compute the value.

$26 \times 25 = 650$ $650 \times 24 = 15,600$ $15,600 \times 23 = 358,800$

Final Answer:

The total number of unique passwords is 358,800.

Would you like further explanation or have any questions? 😊

Related Questions:

1. What would the answer be if repetition of letters was allowed?
2. How does the calculation change if the password length is increased to five characters without repetition?
3. What if only vowels are allowed for constructing the password (a, e, i, o, u)?
4. How many passwords are possible if at least one character must be a digit (0–9)?
5. What is the difference in total combinations if uppercase letters are also included?

Tip:

For problems involving combinations or permutations, carefully identify if repetition is allowed and the total number of choices for each position!