Calculating the Probability of Randomly Selecting a Coded Item with Specific Criteria

In this detailed article, we will break down the process of calculating the probability of randomly selecting a coded item from a catalog where each item is uniquely identified by a specific format: 3 distinct letters followed by 4 distinct nonzero digits, and the first letter must be a vowel while the last digit an even number.

Understanding the Structure of Coded Items

First, let's define the structure of the coded items:

3 Distinct Letters: These letters will start the code and must be distinct and non-repeating. 1st Letter is a Vowel: Out of the 26 letters in the English alphabet, 5 are vowels (A, E, I, O, U). 4 Distinct Nonzero Digits: These digits will follow the letters and must be distinct and range from 1 to 9. Last Digit is Even: The digits 2, 4, 6, and 8 are even.

Step 1: Identifying Possible Vowels and Even Digits

Vowels: There are 5 possible vowels (A, E, I, O, U). Even Digits: There are 4 even digits from 1 to 9 (2, 4, 6, 8).

Step 2: Calculating the Total Number of Coded Items

To find the total number of coded items, we need to calculate the combinations of letters and digits.

Choosing the First Letter (a Vowel): 5 choices. Choosing the Remaining 2 Letters: From the remaining 25 letters (since we cannot reuse the first letter). Total Combinations for Letters: 5 * (25, 2) * 2! 3000. Choosing the 4 Nonzero Digits: From 1 to 9, which gives us (9, 4) * 4! 3024. Total Combinations for Coded Items: 3000 * 3024 9,072,000.

Step 3: Coded Items that Meet the Criteria

We need to find the number of coded items that start with a vowel and end with an even digit.

Choosing the First Letter (a Vowel): 5 choices. Choosing the Remaining 2 Letters: 25 choices for the first and 24 for the second (3000 combinations). Choosing the Last Even Digit: 1 of 4 even digits 4 choices. Choosing the First 3 Digits from Remaining 8 Digits: (8, 3) * 3! 336. Total Coded Items Meeting Criteria: 3000 * 4 * 336 4,032,000.

Step 4: Calculate the Probability

The probability of randomly selecting a coded item that meets the criteria is given by the ratio of favorable outcomes to total outcomes.

Probability: 4,032,000 / 9,072,000 4/9.

Final Answer

The probability of randomly selecting one of these coded items with the first letter as a vowel and the last digit even is 4/9.