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

Assuming there are 26 letters (no distinction among case) and 10 digits:

A) 26 * 26 * 10 = 6760
B) 26 * 25 * 10 = 6500
C) 26 * 1 * 10 = 260

a) Codes

The way of labeling a code with two letters followed by a digit
Assigning one of the 26 letters: A, B, C, D…., F
Assigning one of the 10 digits: 0, 1, 2…, 9
There are two letters so 26
By product rule there are 26×26×10 = 6760 codes

b) Codes with distinct letters
The way the distinct letters are label
25 character of first letters and 25 characters for the second letter so
There are 26×25×10=6500 codes

c) Codes with same letters
The way the same letters are label
26×1×10=260 codes

(a) Ah, the wonderful world of codes! To find the number of codes, we need to consider the possibilities for each element.

For the first letter, we have 26 options (since there are 26 letters in the English alphabet). The second letter also gives us 26 options. Finally, the digit can be any of the 10 digits from 0 to 9.

So, the total number of codes is 26 * 26 * 10 = 6,760.

(b) Now we get to make things a bit more interesting by adding the condition of distinct letters. The first letter still gives us 26 options, but the second letter will have 25 options (since it must be different from the first). The digit, as before, can be any of the 10 digits.

The number of codes with distinct letters is 26 * 25 * 10 = 6,500.

(c) Ah, now we experience the joys of repetition! If we want the same letters, we have no choice but to pick the same letter twice. So, the first letter gives us 26 options, and since the second letter must be the same, it also has only 1 option. The digit can still be any of the 10 digits.

The number of codes with the same letters is 26 * 1 * 10 = 260.

Voilà! We have solved the mystery of the codes.

To find the number of codes, we need to determine the number of possibilities for each part of the code.

(a) Codes:

First, let's consider the two letters. There are 26 letters in the English alphabet, so there are 26 choices for the first letter and 26 choices for the second letter. Since the letters can be the same or different, each letter has the same number of possibilities.

Next, let's consider the digit. There are 10 digits (0-9) to choose from.

Multiplying the number of possibilities for each part together, the total number of codes can be found by:

Number of codes = Number of choices for first letter * Number of choices for second letter * Number of choices for the digit

Number of codes = 26 * 26 * 10 = 6,760

Therefore, there are 6,760 possible codes.

(b) Codes with distinct letters:

In this case, we need to choose two different letters for the first and second positions.

Number of codes with distinct letters = Number of choices for the first letter * Number of choices for the second letter (excluding the first chosen letter) * Number of choices for the digit

Number of codes with distinct letters = 26 * 25 * 10 = 6,500

There are 6,500 possible codes with distinct letters.

(c) Codes with the same letters:

In this case, we need to choose the same letter for the first and second positions.

Number of codes with the same letters = Number of choices for the letter * Number of choices for the digit

Number of codes with the same letters = 26 * 10 = 260

There are 260 possible codes with the same letters.