Givens 4 black marbles,2 white marbles and one red marble determine how many ways all seven marbles can be arranged on a reversible ring calculate both when half of it is symmetric and not symmetric and the add

To solve this problem, we need to determine the number of ways to arrange 7 marbles (4 black, 2 white, and 1 red) on a reversible ring. We will calculate both cases: when the arrangement is symmetric and when it is not symmetric.

### Understanding the Problem
1. **Reversible Ring:** A ring is reversible if flipping it over gives the same arrangement. This means that for each arrangement, we consider it equivalent to its reverse.
2. **Symmetric and Non-Symmetric Cases:**
   - **Symmetric Arrangement:** An arrangement that looks the same when rotated halfway (180 degrees) around the ring.
   - **Non-Symmetric Arrangement:** An arrangement that does not have this symmetry property.

### Step-by-Step Solution

#### Step 1: Calculate Total Number of Arrangements Without Considering Symmetry

Since we have 7 marbles in total, including 4 black (B), 2 white (W), and 1 red (R), the total number of linear arrangements without considering any symmetry is calculated by:

\[
\frac{7!}{4! \cdot 2! \cdot 1!} = \frac{5040}{24 \cdot 2 \cdot 1} = \frac{5040}{48} = 105
\]

#### Step 2: Calculate Arrangements for a Reversible Ring

For a reversible ring, any arrangement is equivalent to its reverse. Therefore, we need to divide the total number of linear arrangements by 2 to account for the reversibility:

\[
\text{Number of arrangements on a reversible ring} = \frac{105}{2} = 52.5
\]

Since the number of ways must be an integer, there are actually two cases to consider:
1. **Symmetric Arrangements:** The arrangements that look the same when flipped.
2. **Non-Symmetric Arrangements:** The arrangements that do not look the same when flipped.

Let:
- \( x \) be the number of symmetric arrangements.
- \( y \) be the number of non-symmetric arrangements.

Thus, we have:

\[
x + y = 105
\]
and for a reversible ring,

\[
x + \frac{y}{2} = 52.5
\]

#### Step 3: Solving the Equations

We can solve these two equations:

1. Multiply the second equation by 2:

\[
2x + y = 105
\]

2. Subtract the first equation from this:

\[
2x + y - (x + y) = 105 - 105
\]
\[
x = 35
\]

So, there are 35 symmetric arrangements. Now substitute \( x = 35 \) back into the first equation:

\[
35 + y = 105
\]
\[
y = 70
\]

Thus, there are 70 non-symmetric arrangements.

#### Step 4: Add the Two Cases Together

Finally, to find the total number of ways to arrange the marbles on a reversible ring considering both symmetric and non-symmetric cases, we add these together:

\[
\text{Total number of arrangements} = x + y = 35 + 70 = 105
\]

### Conclusion

- **Symmetric arrangements:** 35
- **Non-symmetric arrangements:** 70
- **Total arrangements on a reversible ring:** 105

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

### Related Questions

1. What would be the number of arrangements if the ring was not reversible?
2. How does the number of ways change if all the marbles were of the same color?
3. Can you explain the concept of "rotational symmetry" and how it affects counting arrangements?
4. How would the calculation change if there were 3 white marbles instead of 2?
5. Can this problem be generalized to more marbles or different objects?

#### Tip

When counting arrangements on a ring, always consider rotational and reflectional symmetries to avoid overcounting.

Given 4 black marbles, 2 white marbles, and 1 red marble, determine how many ways all seven marbles can be arranged on a reversible ring. Calculate both when half of it is symmetric and not symmetric, and then add the results.

Explore how to arrange 4 black, 2 white, and 1 red marble on a reversible ring, accounting for symmetry and non-symmetry. This combinatorics problem uses permutation and symmetry principles, ideal for high school students.

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