The number of ways in which four boys can be seated around a round table in four chairs of different colours, is

Options

A. `""^(16)C_(11)`
B. `""^(16)C_(5)`
C. `""^(16)C_(9)`
D. `""^(20)C_(8)`

Answer

To find the number of ways in which four boys can be seated around a round table with four chairs of different colors, we can follow these steps:

### Step 1: Understand the Arrangement
Since the chairs are of different colors, the arrangement of the boys matters. We can treat the arrangement as a linear arrangement first, and then adjust for the circular nature of the table.

### Step 2: Fix One Boy
In circular permutations, we can fix one boy in one chair to eliminate the effect of rotations. This means we will consider the arrangement of the remaining boys around the table.

### Step 3: Arrange the Remaining Boys
After fixing one boy, we have three boys left to arrange. The number of ways to arrange these three boys in the remaining chairs is given by the factorial of the number of boys left, which is \(3!\).

### Step 4: Calculate the Factorial
Now, we calculate \(3!\):
\[ 3! = 3 \times 2 \times 1 = 6 \]

### Step 5: Consider the Different Chair Colors
Since the chairs are of different colors, each arrangement of boys corresponds to a unique arrangement of colored chairs. Therefore, we multiply the number of arrangements by the number of ways to assign colors to the chairs.

### Step 6: Final Calculation
Since we have four different colored chairs, we can assign the colors in \(4!\) ways:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 7: Total Arrangements
Now, we multiply the arrangements of boys by the arrangements of chair colors:
\[ \text{Total arrangements} = 3! \times 4! = 6 \times 24 = 144 \]

### Final Answer
Thus, the total number of ways in which four boys can be seated around a round table in four chairs of different colors is **144**.

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