A total of 5 number cards from 1 to 5 are placed in order, assuming that the number
cards 2 and 3 are taken out, then placed randomly but must be next to each other. How
many different ways to place the cards in total?

To solve this problem, we need to consider how we can arrange the 5 cards where cards 2 and 3 are taken out and then placed back next to each other in a random order.

### Step-by-Step Breakdown:

1. **Treat the pair (2 and 3) as a single block**: Since cards 2 and 3 must be next to each other, we can treat the pair as one "block." There are two possible internal arrangements of this block:
   - (2, 3)
   - (3, 2)

2. **Number of objects to arrange**: 
   After treating the pair of cards as one block, you now have four objects to arrange: the block (2-3), and the remaining cards 1, 4, and 5.

3. **Arranging the four objects**: 
   The four objects (1, block, 4, 5) can be arranged in \(4!\) ways (since the order of 4 distinct objects can be rearranged in factorial of 4).
   \[
   4! = 24
   \]

4. **Arranging the cards within the block**: 
   As mentioned earlier, there are two ways to arrange the cards within the block:
   - Card 2 first, then card 3.
   - Card 3 first, then card 2.
   
   Thus, there are 2 possible arrangements for the internal order of the block.

5. **Total number of arrangements**:
   The total number of different ways to arrange the cards is the product of the number of ways to arrange the 4 objects and the number of ways to arrange the two cards in the block:
   \[
   24 \times 2 = 48
   \]

### Final Answer:
There are **48 different ways** to place the cards in total.

---

Would you like more details or further clarification? Here are some related questions to expand on this concept:

1. How would the result change if card 2 and card 4 had to be adjacent instead?
2. What if cards 2, 3, and 4 were removed and had to stay adjacent in any order?
3. How would the problem change if card 3 could not be placed next to card 4?
4. If we wanted to count the number of ways to place only two of the remaining cards, how many ways would there be?
5. What is the total number of arrangements if no restrictions are placed on any cards?

**Tip**: When solving problems involving adjacent blocks, grouping them together as one "object" simplifies the process.

A total of 5 number cards from 1 to 5 are placed in order, assuming that the number cards 2 and 3 are taken out, then placed randomly but must be next to each other. How many different ways to place the cards in total?

Discover how to solve a combinatorics problem where 5 number cards are arranged, but cards 2 and 3 must be adjacent. Learn key mathematical concepts like permutations and factorial calculations in this step-by-step solution. Suitable for high school 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