Question

a password is 4 characters long and must consist of 3 letters and one number. if letters cannot be repeated and the password must end with a number, how many possibilities are there?
a. 175,760
b. 158,184
c. 156,000
d. 140,400
please select the best answer from the choices provided
a
b
c
d

Explanation:

Step1: Analyze the password structure

The password has 4 characters: 3 letters (no repetition) and 1 number (at the end). First, calculate the number of ways to choose and arrange the 3 non - repeating letters. There are 26 letters in the alphabet. The number of permutations of 26 letters taken 3 at a time is given by the formula \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 26\) and \(r=3\). So \(P(26,3)=\frac{26!}{(26 - 3)!}=\frac{26!}{23!}=26\times25\times24\).

Step2: Calculate the number of letter permutations

Calculate \(26\times25\times24 = 15600\).

Step3: Analyze the number part

For the number at the end, we assume it is a single - digit number (0 - 9), so there are 10 possible choices.

Step4: Calculate the total number of passwords

To find the total number of possible passwords, we multiply the number of letter arrangements by the number of number choices. So the total number of passwords is \(15600\times10=156000\).

Answer:

c. 156,000