A 4-letter code word consists of letters A, B, and C. If the code includes all the three letters, how many such codes are possible?

A 4-letter code word consists of letters A, B, and C. If the code includes all the three letters, how many such codes are possible?

We need to form a 4-letter code using A, B, and C, and it must include all three letters.

Let's consider the arrangements of the letters:
We must have one of the letters repeated.

Case 1: The repeated letter is A.
Possible arrangements: AABC, AACB, ABAA, ABCA, ACAA, AABC, ACAB, ACBA

Case 2: The repeated letter is B.
Possible arrangements: BABC, BACB, BBAA, BBCA, BCAA, BABC, BCAB, BCBA

Case 3: The repeated letter is C.
Possible arrangements: CABC, CACB, CBAA, CBCA, CBAA, CABC, CABCA, CBAB, CCBA

Each case has 4!/2! = 12 arrangements.

So, there are 3 * 12 = 36 possible codes.

Thus, there are 36 different 4-letter codes that include all three letters A, B, and C.