Suppose a code consist of two letters followed by a digit. Find the number of

(a) Codes
(b) Codes with distinct letters
(c) Codes with the same letters.

the answer of a is 26*26*10*10*10.

the answer of b is 26*25*10*10*10.
of c is 26*1*10*10*10

The number of possible different arrangements would be 26 x 26 x 10 = 6760

To find the number of codes in the given format, we will consider two cases separately: one with distinct letters and one with the same letters.

Case 1: Codes with distinct letters
(a) To find the number of possible codes, we need to determine the number of choices for each position (letter and digit).

For the first letter, we have 26 choices (any letter of the alphabet).
For the second letter, since it needs to be distinct from the first letter, we have 25 choices remaining.
For the digit, we have 10 choices (0-9).

To calculate the total number of codes, we multiply the choices for each position: 26 * 25 * 10 = 6,500.

Therefore, there are 6,500 codes with distinct letters.

(b) If we want to count only the codes with distinct letters, we don't have to consider the digit as it is the same for all codes. We only need to calculate the number of choices for the two letter positions.

For the first letter, we have 26 choices (any letter of the alphabet).
For the second letter, as it needs to be distinct from the first letter, we have 25 choices remaining.

To calculate the total number of codes with distinct letters, we multiply the choices for each position: 26 * 25 = 650.

Therefore, there are 650 codes with distinct letters.

Case 2: Codes with the same letters
(c) To find the number of codes with the same letters, we consider the number of choices for each position (letter and digit).

For the first letter, we have 26 choices (any letter of the alphabet).
Since the second letter needs to be the same as the first letter, we have no choice. It is fixed once the first letter is chosen.
For the digit, we have 10 choices (0-9).

To calculate the total number of codes with the same letters, we multiply the choices for each position: 26 * 1 * 10 = 260.

Therefore, there are 260 codes with the same letters.

In summary:
(a) Total number of codes: 6,500.
(b) Codes with distinct letters: 650.
(c) Codes with the same letters: 260.