Consider a collection of cards numbered from 1 to 9. 1 23 4 5 6 789 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. An example is shown below. 1 8 4 3 5 7 26 9 (a) How many ways can this be done? 


We are given a collection of cards numbered from 1 to 9. The odd numbers are \(1, 3, 5, 7, 9\), and they must be arranged in increasing order from left to right. The even-numbered cards, which are \(2, 4, 6, 8\), can be placed in any order.

### Key Observations:
- The odd numbers \(1, 3, 5, 7, 9\) must always appear in increasing order, so their relative positions are fixed.
- The even numbers \(2, 4, 6, 8\) can be arranged in any order in the remaining positions.

### Step-by-Step Solution:

1. **Choosing the positions for the odd numbers**:  
   There are 5 odd numbers, so we must choose 5 positions from the 9 available spots for these numbers. The number of ways to choose 5 positions from 9 is given by the combination formula:
   \[
   \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9 \times 8 \times 7 \times 6 \times 5!}{5!} = 126
   \]

2. **Arranging the even numbers**:  
   Once the positions for the odd numbers are fixed, the remaining 4 positions will be filled by the even numbers \(2, 4, 6, 8\). These can be arranged in any order, so there are \(4!\) ways to arrange them:
   \[
   4! = 4 \times 3 \times 2 \times 1 = 24
   \]

3. **Total number of ways**:  
   To find the total number of ways to arrange the cards, we multiply the number of ways to choose the positions for the odd numbers by the number of ways to arrange the even numbers:
   \[
   \text{Total ways} = \binom{9}{5} \times 4! = 126 \times 24 = 3024
   \]

Thus, the total number of ways to arrange the cards is \( \boxed{3024} \).

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Would you like more details, or do you have any questions?

Here are 5 follow-up questions for further exploration:
1. How would the problem change if the odd numbers could also be arranged in any order?
2. What if we added one more card, making it a 10-card sequence?
3. How would the solution change if the even numbers also had to be arranged in increasing order?
4. What if we wanted the odd numbers to be in decreasing order instead of increasing?
5. Can you extend this approach to a set of numbers from 1 to 11?

**Tip**: When arranging elements with fixed order constraints, always focus first on positioning the constrained elements before arranging the others!

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. How many ways can this be done?

Learn how to arrange cards numbered from 1 to 9, where odd numbers are fixed in increasing order, and even numbers can be arranged freely. This combinatorics problem uses the combination and permutation principles, with a total of 3024 possible arrangements. Suitable for students in Grades 9-12.

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