Question

a password is 4 characters long and must consist of 3 letters and 1 of 10 special characters. if letters can be repeated and the special character is at the end of the password, how many possibilities are there?
a. 175,760
b. 456,976
c. 703,040
d. 1,679,616
please select the best answer from the choices provided
a
b
c
d

Explanation:

Step1: Determine the number of choices for each part

• For the 3 letters (no repetition, 26 letters in total): The number of permutations of 26 letters taken 3 at a time is \( P(26, 3)=\frac{26!}{(26 - 3)!}=26\times25\times24\).
• For the 1 special character (10 special characters, and it's at the end, so we choose 1 out of 10, but since it's fixed at the end, the number of choices for the special character position is 10 (but we will handle the order later). Wait, actually, the password structure: 3 letters, 1 special character (at the end), and we need to consider the number of ways to arrange? Wait, no, the password is 4 characters: 3 letters (no repetition) and 1 special character (at the end). Wait, the problem says "consist of 3 letters and 1 of 10 special characters. If letters can be repeated? Wait, no, the original problem: "If letters can be repeated and the special character is at the end of the password, how many possibilities are there?" Wait, let's re - read: "A password is 4 characters long and must consist of 3 letters and 1 of 10 special characters. If letters can be repeated and the special character is at the end of the password, how many possibilities are there?"

So, for each of the 3 letter positions:

• Since letters can be repeated, each letter position has 26 choices. So for 3 letter positions, the number of ways is \(26\times26\times26 = 26^{3}\).
• For the special character (at the end), there are 10 choices.

Step2: Calculate the total number of possibilities

The total number of passwords is the product of the number of ways to choose the letters and the number of ways to choose the special character.

So total possibilities \(=26^{3}\times10\)

First, calculate \(26^{3}=26\times26\times26 = 17576\)

Then, \(17576\times10 = 175760\)

Wait, but this is option a. But wait, maybe I misread the problem. Wait, the problem says "3 letters and 1 of 10 special characters" and "4 characters long". Wait, maybe the letters can be repeated, and the special character is at the end. So 3 letters (each with 26 options, since repetition is allowed) and 1 special character (10 options). So \(26\times26\times26\times10=26^{3}\times10 = 17576\times10 = 175760\)

Answer:

a. 175,760