A security code consists of four characters generated in order: a letter, a digit, a lette… — ACT

A security code consists of four characters generated in order: a letter, a digit, a letter, and a digit.

- The 1st character is a letter chosen from the 5-element set $\{A, B, C, D, E\}$ .
- The 2nd character is a digit chosen from the 9 non-zero digits $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ .
- The 3rd character is a letter chosen from $\{A, B, C, D, E\}$ , but it cannot be the same as the 1st character.
- The 4th character is an even digit chosen from $\{2, 4, 6, 8\}$ , but it cannot be the same as the 2nd character.

How many different security codes can be formed following these rules?

Answer: 640

Answer

640

The problem requires counting combinations with dependent restrictions, which can be solved by splitting into two mutually exclusive cases. If the 2nd character is an odd digit (5 possibilities), the 4th character has all 4 even digits available, giving 5 × 5 × 4 × 4 = 400 valid codes. If the 2nd character is an even digit (4 possibilities), the 4th character cannot match it, leaving 3 options, giving 5 × 4 × 4 × 3 = 240 valid codes. Summing the two distinct cases gives 400 + 240 = 640.

Step-by-Step Solution

Analyze the conditions for the digit positions (2nd and 4th characters).

The restrictions require dividing the counting into two cases based on the parity of the 2nd character.

The 4th character must be chosen from the 4 even digits {2, 4, 6, 8}. If the 2nd character is an odd digit, it does not reduce the choices for the 4th character. If the 2nd character is an even digit, it reduces the remaining choices for the 4th character from 4 to 3.

Calculate total possibilities for Case 1 (2nd character is odd).

5 × 5 × 4 × 4 = 400 codes.

There are 5 options for the 1st letter, 5 odd digits {1, 3, 5, 7, 9} for the 2nd character, 4 remaining letters for the 3rd character, and 4 even digits for the 4th character.

Calculate total possibilities for Case 2 (2nd character is even).

5 × 4 × 4 × 3 = 240 codes.

There are 5 options for the 1st letter, 4 even digits {2, 4, 6, 8} for the 2nd character, 4 remaining letters for the 3rd character, and 3 remaining even digits for the 4th character.

Sum the results of the two cases.

400 + 240 = 640 security codes.

According to the Addition Principle of counting, the total number of outcomes for mutually exclusive events is the sum of their individual outcomes.

Key Concept

Fundamental Counting Principle with Case Analysis

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