Find the total number of ways, in which 10 beads can be strung into a necklace?

Answer

To find the total number of ways to string 10 beads into a necklace, we can follow these steps:

### Step 1: Understand the arrangement of beads in a necklace
When arranging beads in a necklace, we need to consider that the arrangement is circular. This means that rotating the necklace does not create a new arrangement.

### Step 2: Use the formula for circular arrangements
For circular arrangements, the formula is given by (n-1)!, where n is the number of beads. Here, n = 10, so we will use (10-1)! = 9!.

### Step 3: Account for the reversibility of the necklace
Since a necklace can be flipped over and still look the same, we need to divide the total arrangements by 2. Therefore, the formula for the number of distinct arrangements of beads in a necklace is:
\[ \text{Number of arrangements} = \frac{(n-1)!}{2} \]

### Step 4: Substitute the value of n
Now, substituting n = 10 into the formula:
\[ \text{Number of arrangements} = \frac{(10-1)!}{2} = \frac{9!}{2} \]

### Step 5: Calculate 9!
Now, we calculate 9!:
\[ 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880 \]

### Step 6: Divide by 2
Now we divide 9! by 2:
\[ \frac{9!}{2} = \frac{362880}{2} = 181440 \]

### Final Answer
Thus, the total number of ways in which 10 beads can be strung into a necklace is **181440**.

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