https://file.aiquickdraw.com/mathbot/file/172596503547556rx25ff.jpg

Let's analyze the problem step by step:

### Given:
- Cards numbered 1 to 9.
- The cards must be arranged in a row.
- Odd-numbered cards (1, 3, 5, 7, 9) must be arranged in increasing order from left to right.
- Even-numbered cards (2, 4, 6, 8) can be arranged in any order.

### Part (a): 
**How many ways can this be done?**

### Solution:

1. **Odd-numbered cards**:
   - The odd-numbered cards \(1, 3, 5, 7, 9\) are fixed in increasing order. So their positions are restricted, and we cannot change their order.

2. **Even-numbered cards**:
   - The even-numbered cards \(2, 4, 6, 8\) can be placed in any of the four remaining positions.
   - The number of ways to arrange 4 even cards in 4 available positions is the number of permutations of 4 objects, which is \(4!\).

3. **Calculation**:
   \[
   4! = 4 \times 3 \times 2 \times 1 = 24
   \]

Thus, the total number of ways to arrange the cards is **24**.

Would you like further details or have any questions? Here are some related questions to explore:

1. What if the even-numbered cards were also restricted to a specific order?
2. How many ways can the cards be arranged if both the odd and even cards had no restrictions?
3. How would the number of arrangements change if we had more than 9 cards?
4. What if the odd-numbered cards were arranged in decreasing order instead of increasing?
5. Can you generalize this type of arrangement problem for other sets of numbers?

**Tip:** When dealing with problems involving restricted orders (like the odd numbers here), always isolate the fixed elements first, then focus on the flexible parts (like the even numbers in this case).

Consider a collection of cards numbered from 1 to 9. All nine cards are arranged in a row so that if the numbers are read from left to right, the odd numbers are in increasing order. The even-numbered cards can be in any order. (a) How many ways can this be done?

Learn how to arrange cards numbered from 1 to 9 where the odd numbers must be in increasing order, and the even numbers can be arranged in any order. This problem involves key concepts of permutations and combinatorics. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation