https://file.aiquickdraw.com/mathbot/file/17290554102025gn4olqg.jpg

The problem in the image asks to calculate how many distinct identification numbers are possible, given certain rules:

- The identification number consists of two letters, followed by three digits, and then three more letters.
- Repetitions are not allowed within any of the three groups.
- However, letters from the first group of two may occur again in the last group of three.

Let's break down the problem:

1. **First group of two letters**:
   - There are 26 letters in the alphabet.
   - Since repetition is not allowed, for the first letter, we have 26 choices.
   - For the second letter, there are 25 choices (since it cannot be the same as the first).

   So, the number of possible combinations for the first two letters is:
   \[
   26 \times 25 = 650
   \]

2. **Three digits**:
   - There are 10 digits (0-9).
   - For the first digit, there are 10 choices.
   - For the second digit, 9 choices (since it cannot be the same as the first).
   - For the third digit, 8 choices (since it cannot be the same as either of the first two).

   The number of combinations for the digits is:
   \[
   10 \times 9 \times 8 = 720
   \]

3. **Last group of three letters**:
   - Letters from the first group can appear again in this group, but within this group, repetition is not allowed.
   - For the first letter, we still have 26 choices (since letters from the first group can be reused).
   - For the second letter, we have 25 choices (since it cannot be the same as the first letter in this group).
   - For the third letter, there are 24 choices.

   The number of combinations for the last three letters is:
   \[
   26 \times 25 \times 24 = 15,600
   \]

### Final calculation:
To find the total number of distinct identification numbers, we multiply the results from all three groups:
\[
650 \times 720 \times 15,600 = 7,300,800,000
\]

So, the total number of distinct identification numbers possible is **7,300,800,000**.

Let me know if you'd like further clarification or more details! 

Here are 5 related questions:
1. What happens to the total number if repetitions were allowed in all groups?
2. How would the count change if letters and digits could not repeat at all across the entire sequence?
3. What is the probability of randomly generating a valid identification number under these rules?
4. If the first group could only use vowels, how would that affect the total possibilities?
5. What would the count be if each digit in the middle could only be odd?

**Tip:** When working with permutations, always carefully check whether repetition is allowed and in which groups.

Subject identification numbers in a certain scientific research project consist of two letters followed by three digits and then three more letters. Assume repetitions are not allowed within any of the three groups, but letters in the first group of two may occur also in the last group of three. How many distinct identification numbers are possible?

Explore how to calculate the total number of distinct subject identification numbers, consisting of letters and digits with specific repetition rules. This problem covers combinatorial concepts, including permutations without repetition, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation