By stringing together 9 different coloured beads, how many different bracelets can be made

Options

A. 20160
B. 40320
C. 80640
D. 10080

Answer

To solve the problem of how many different bracelets can be made by stringing together 9 different colored beads, we can follow these steps:

### Step 1: Understand the arrangement of beads
When arranging beads in a circle, the arrangements are considered the same if they can be rotated into one another. For n different colored beads, the number of arrangements in a circle is given by (n - 1)!.

### Step 2: Calculate the arrangements for 9 beads
In this case, we have 9 different colored beads. Therefore, the number of arrangements in a circle is:
\[ (9 - 1)! = 8! \]

### Step 3: Calculate 8!
Now, we need to calculate 8! (8 factorial):
\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \]

### Step 4: Account for the bracelet's symmetry
Since a bracelet can be flipped (i.e., it has symmetry), arrangements that are mirror images of each other are considered the same. Therefore, we need to divide the total arrangements by 2:
\[ \text{Number of distinct bracelets} = \frac{8!}{2} = \frac{40320}{2} = 20160 \]

### Final Answer
Thus, the total number of different bracelets that can be made with 9 different colored beads is:
\[ \boxed{20160} \]

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