An e-mail password must contain three characters. The password has to contain one numeral from 0 to 9, one upper case and one lower case character from the English alphabet. How many distinct passwords are possible ?

Question

Accepted Answer

To determine the number of distinct passwords possible under the given conditions, we need to consider the requirements for the password: it must contain one numeral, one uppercase letter, and one lowercase letter. Let's break down the calculation:

1. **Numerals (0-9):** There are 10 possible choices for the numeral.

2. **Uppercase Letters (A-Z):** There are 26 possible choices for the uppercase letter.

3. **Lowercase Letters (a-z):** There are 26 possible choices for the lowercase letter.

Since the password must contain one numeral, one uppercase letter, and one lowercase letter, we can calculate the total number of combinations by multiplying the number of choices for each component:

$$ 
10 \, (\text{numerals}) \times 26 \, (\text{uppercase letters}) \times 26 \, (\text{lowercase letters}) = 6760 
$$

However, this calculation only accounts for the number of ways to choose the characters, not the order in which they can be arranged. Since the password consists of three characters, and each character can be in any of the three positions, we must also consider the permutations of these three characters:

$$ 
3! = 6 
$$

Thus, the total number of distinct passwords is:

$$ 
6760 \times 6 = 40560 
$$

This calculation confirms that Option 3, which states there are 40560 distinct passwords possible, is correct.

**Explanation for Incorrect Options:**

- **Option 1 (6760):** This option only considers the number of ways to choose the characters without accounting for the different permutations of these characters in the password. It fails to multiply by the number of possible arrangements (3!), which is why it is incorrect.

- **Option 2 (13520):** This option does not align with any logical breakdown of the problem. It neither represents the number of ways to choose the characters nor the permutations correctly, making it incorrect.

- **Option 4 (105456):** This option overestimates the number of possible passwords. It likely results from an incorrect calculation or misunderstanding of the permutations and combinations involved, making it incorrect.

Therefore, based on the detailed breakdown and logical reasoning, Option 3 is the correct answer.