In how many ways can 8 boys be seated at a round table?

Answer

To find the number of ways to seat 8 boys at a round table, we can follow these steps:

### Step 1: Understand the Circular Arrangement
In circular arrangements, one position is typically fixed to eliminate the effect of rotations. This is because rotating the entire arrangement does not create a new unique arrangement.

### Step 2: Calculate the Number of Arrangements
For 'n' objects seated in a circle, the formula to calculate the arrangements is given by:
\[ (n - 1)! \]
In this case, we have 8 boys, so we substitute \( n = 8 \):
\[ (8 - 1)! = 7! \]

### Step 3: Calculate \( 7! \)
Now, we need to calculate \( 7! \):
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]
Calculating this step-by-step:
- \( 7 \times 6 = 42 \)
- \( 42 \times 5 = 210 \)
- \( 210 \times 4 = 840 \)
- \( 840 \times 3 = 2520 \)
- \( 2520 \times 2 = 5040 \)
- \( 5040 \times 1 = 5040 \)

Thus, \( 7! = 5040 \).

### Final Answer
Therefore, the total number of ways to arrange 8 boys at a round table is:
\[ \text{Total arrangements} = 5040 \]

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