How Many Different Necklaces Can Be Made Using 7 Black Beads, 1 Red Bead, and 1 Green Bead?

Introduction to Burnside's Lemma

Burnside's Lemma is a powerful tool in combinatorial mathematics that helps us count distinct arrangements considering symmetries, rotations, and reflections. In the context of this article, we'll use it to determine the number of unique necklaces that can be made using 7 black beads, 1 red bead, and 1 green bead.

Step-by-Step Analysis Using Burnside's Lemma

To apply Burnside's Lemma, we'll follow these steps: Calculate Total Beads:

We have a total of n 9 beads, including 7 black beads, 1 red bead, and 1 green bead.

Count Rotational Symmetries:

For a necklace of 9 beads, there are 9 rotational symmetries: 0 through 8 positions.

Count Fixed Arrangements Under Rotations:

We will examine how many arrangements remain unchanged for each possible rotation.

Calculate Total Fixed Arrangements:

We will summarize the fixed arrangements for each rotation.

Apply Burnside's Lemma:

Finally, we'll calculate the average number of fixed arrangements across all symmetries.

Step 1: Calculate Total Beads

We have n 9 beads in total, with 7 black beads, 1 red bead, and 1 green bead.

Step 2: Count Rotational Symmetries

For a necklace of 9 beads, there are 9 rotational symmetries: 0 through 8 positions.

Step 3: Count Fixed Arrangements Under Rotations

We will examine how many arrangements remain unchanged for each possible rotation. Rotation by 0 positions (identity rotation):

All arrangements are fixed. The total number of arrangements can be calculated using the multinomial coefficient:

frac{9!}{7! cdot 1! cdot 1!}  frac{9 times 8}{1 times 1}  72

Rotation by 1, 2, 4, 5, 7, and 8 positions:

No arrangements can remain unchanged because the black beads would need to occupy the same positions as the red and green beads, which is impossible given their counts.

Rotation by 3 positions:

The arrangement must be periodic with a period of 3. This means we can group the beads into 3 sets of 3.

For the black beads, we need to have 3 black beads in each group, but we only have 7 black beads, which is not possible. Therefore, there are 0 fixed arrangements.

Rotation by 6 positions:

Similarly, the arrangement must be periodic with a period of 3, leading to the same conclusion as the 3-position rotation. Thus, there are 0 fixed arrangements.

Step 4: Calculate Total Fixed Arrangements

Summarizing the fixed arrangements: Rotation by 0 positions: 72 Rotation by 1, 2, 4, 5, 7, and 8 positions: 0 Rotation by 3 positions: 0 Rotation by 6 positions: 0 Total fixed arrangements 72 0 0 0 0 0 0 0 0 72

Step 5: Apply Burnside's Lemma

The average number of fixed arrangements across all symmetries is:

72 / 9 8

Conclusion

Thus, the total number of different necklaces that can be made with 7 black beads, 1 red bead, and 1 green bead is 8.