Question

a password is 4 characters long, consisting of 2 letters and 2 numbers. the password must begin and end with a letter. if numbers can be repeated but letters cannot, how many possibilities are there?
a. 65,000
b. 60,840
c. 58,500
d. 52,650
please select the best answer from the choices provided.
a
b
c
d

Explanation:

Step1: Determine letter choices

There are 26 letters (A - Z). For the first and last characters (letters), since letters can't repeat, the first letter has 26 choices, the last letter has 25 choices.

Step2: Determine number choices

There are 10 digits (0 - 9). For the two middle characters (numbers), numbers can repeat, so each number has 10 choices. So number permutations: \(10\times10 = 100\).

Step3: Calculate total possibilities

Multiply letter permutations and number permutations: \(26\times25\times100=65000\)? Wait, no, wait the password is 4 characters: first (letter), second (number), third (number), fourth (letter). So first letter: 26, second number:10, third number:10, fourth letter:25 (since can't repeat first letter). So total: \(26\times10\times10\times25 = 26\times25\times100=65000\)? Wait but wait, maybe I misread the structure. Wait the problem says "consisting of 2 letters and 2 numbers. The password must begin and end with a letter." So structure: Letter, Number, Number, Letter. So first letter: 26 options, second number:10, third number:10, fourth letter:25 (since can't repeat the first letter). So total combinations: \(26\times10\times10\times25 = 26\times25\times100 = 65000\)? Wait but let's recalculate: \(26\times10 = 260\), \(25\times10 = 250\), then \(260\times250 = 65000\). Wait but the options have 65,000 as option a. But wait, maybe I made a mistake. Wait, letters: first letter 26, fourth letter: 25 (no repeat). Numbers: two numbers, each 10 options (since numbers can repeat). So total: \(26\times10\times10\times25 = 26\times25\times100 = 65000\). So the answer should be a. 65,000.

Answer:

a. 65,000