A baby has nine different toy animals. Five of them are red
and four of them are blue. She arranges them in a line so that the colours are arranged symmetrically. How many different arrangements are possible?

To solve this problem, we need to consider the symmetry constraint and the different arrangements of red and blue toys.

### Step 1: Understanding the Symmetry Constraint
Since the arrangement needs to be symmetric, if the baby places a toy in the first position, the corresponding toy at the last position must be of the same color. Similarly, the second toy must match the second-to-last toy, and so on.

For 9 toys, the middle toy (5th position) doesn't have a counterpart, so it can be any of the 9 toys. The other 8 toys will be arranged symmetrically around this middle toy.

### Step 2: Counting Arrangements for the Middle Toy
The middle toy can be any of the 9 toys, so there are 9 choices for the middle toy.

### Step 3: Counting Symmetric Arrangements for the Remaining 8 Toys
Now, we focus on the remaining 8 toys (4 pairs) which must be arranged symmetrically. Out of these 8 toys, 4 are red and 4 are blue.

- We need to choose 2 positions for red toys out of the first 4 positions. This will automatically determine the positions of the blue toys because the arrangement is symmetric.
- The number of ways to choose 2 positions out of 4 is given by the combination formula \(\binom{4}{2}\).

\[
\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6
\]

### Step 4: Total Number of Arrangements
The total number of symmetric arrangements is the product of the number of choices for the middle toy and the number of symmetric arrangements for the remaining 8 toys:

\[
9 \times 6 = 54
\]

### Final Answer:
There are **54 different symmetric arrangements** possible.

---

Do you want more details or have any questions about the solution? Here are 5 related questions for further exploration:

1. What if the baby had 10 toys, with 5 red and 5 blue? How would the symmetry constraint affect the arrangements?
2. How would the solution change if the number of red and blue toys were not equal?
3. What if there were 3 different colors instead of 2? How would that affect the symmetric arrangements?
4. Can you derive a general formula for the number of symmetric arrangements with different numbers of toys?
5. What would be the impact on the solution if the toys were distinguishable by more than just color?

**Tip:** Symmetry problems often reduce the complexity of arrangements by fixing certain elements, so always look for ways to simplify the problem through symmetry constraints.

Explore how to calculate the number of symmetric arrangements of nine toys, consisting of five red and four blue toys. Understand the symmetry constraint and apply combinatorial principles to determine possible arrangements.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation