A vehicle registration number consists of two distinct English letters followed by two digits.\nHow many distinct registration numbers can be formed in this format?

Introduction / Context:
This is a straightforward counting question involving arrangements of letters and digits with a simple restriction on the letters. It checks whether the concept of permutations with and without repetition is correctly applied in a multi part format such as a registration number.

Given Data / Assumptions:

• The registration number structure is: Letter, Letter, Digit, Digit.
• The two letters must be distinct uppercase English letters.
• There are 26 possible letters (A to Z).
• Each digit can be any of 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9.
• Digits can repeat; there is no restriction on them.

Concept / Approach:
We treat each position independently and then apply the multiplication principle. For the letter positions, order matters and letters cannot repeat, so we use permutations without repetition. For the digit positions, order matters but digits may repeat, so we simply count the number of ways for each digit position and multiply. Finally, we multiply the letter choices and digit choices to get the total number of registration numbers.

Step-by-Step Solution:
For the first letter position, there are 26 choices. For the second letter position, one letter has already been used, so there are 25 remaining choices. Thus, the letters can be chosen and arranged in 26 * 25 ways. For each digit position (third and fourth characters), there are 10 choices (0 to 9). Since digits can repeat, the number of ways for the two digit positions is 10 * 10 = 100. Total distinct registration numbers = (26 * 25) * 100. Compute 26 * 25 = 650. Now 650 * 100 = 65000.

Verification / Alternative check:
We can quickly check the magnitude. There are 26 * 25 = 650 ways to pick the letter part, which is less than 26^2 because repetition is not allowed. There are 100 possibilities for the digit part. Multiplying produces 65000, which is less than the 26^2 * 10^2 total that would occur if letters could repeat, so the number appears reasonable and consistent with the restriction.

Why Other Options Are Wrong:

• 64000: This suggests mistakenly using 25 * 25 for the letters, which would wrongly allow the first letter to have only 25 options.
• 72000: This might come from using 26 * 27 or some other incorrect combination for the letters.
• 36000: Too small; likely from 26 * 10 * 10 without the second letter choice or from mixing addition and multiplication.

Common Pitfalls:
A common error is ignoring the condition that the two letters must be distinct, and simply using 26 choices for each, which overcounts some numbers. Another is to forget that digits can repeat and incorrectly treat them like letters with no repetition. Always identify clearly which positions allow repeated symbols and which do not, and then apply permutations or simple counts accordingly.

Final Answer:
The total number of distinct registration numbers possible in this format is 65000.