https://file.aiquickdraw.com/mathbot/file/1729733646858n25x9tta.jpg

Here is a breakdown of the questions from the image:

1. **Problem 1:** There are six digits \(1, 3, 4, 5, 7, 8\) to be arranged into a 3-digit odd number. The total number of odd numbers that can be arranged is:
   - A. 64
   - B. 112
   - C. 120
   - D. 144
   - E. 240

2. **Problem 2:** If mixing two different chemicals results in a new chemical, and there are six distinct chemicals available, how many new chemicals can be formed?
   - A. 12
   - B. 15
   - C. 20
   - D. 24
   - E. 32

3. **Problem 3:** From the digits \(2, 3, 5, 6, 7, 9\), create 3-digit numbers that are less than 400. How many such numbers can be formed?
   - A. 10
   - B. 20
   - C. 40
   - D. 80
   - E. 120

4. **Problem 4:** A school is selecting a student council consisting of a president, vice president, and secretary. If there are 10 candidates, how many ways can the council be chosen?
   - A. 330 ways
   - B. 440 ways
   - C. 620 ways
   - D. 660 ways
   - E. 720 ways

5. **Problem 5:** From the digits \(1, 2, 3, 4, 5, 6\), create 3-digit numbers that are less than 500. How many distinct numbers can be formed?
   - A. 80
   - B. 120
   - C. 150
   - D. 180
   - E. 200

6. **Problem 6:** From a group of 6 people, a team of 4 will be selected: one president, one secretary, and two members. How many different team configurations are possible?
   - A. 30
   - B. 80
   - C. 120
   - D. 210
   - E. 720

Would you like help with specific problems or explanations for how to solve any of them? Here are 5 questions to expand on the current problems:

1. How do you calculate the number of odd numbers formed from given digits?
2. What is the combination formula used to select 2 chemicals from 6?
3. How do restrictions like "less than 400" affect the total number of possibilities?
4. How are permutation principles applied to selecting a student council?
5. What changes if the team selection in problem 6 included different roles?

**Tip:** For any counting problems, always check if order matters (permutations) or if it does not (combinations).

There are six problems involving combinatorics, number formations, and selection of teams. These include tasks like counting how many odd numbers can be formed from given digits and selecting a student council from 10 candidates.

Explore how to calculate permutations and combinations with real-world examples such as selecting student councils and forming numbers under specific constraints. Suitable for Grades 10-12, this content includes step-by-step guides to key concepts like combinations and factorials.

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