There are 3 boys and 4 girls, seated around a circular table so that no two boys are together. Find the number of ways in which this can be done.

Options

A. 36
B. 120
C. 144
D. 132

Answer

To solve the problem of seating 3 boys and 4 girls around a circular table such that no two boys are seated next to each other, we can follow these steps:

### Step 1: Arrange the Girls
Since the arrangement is circular, we can fix one girl to eliminate the effect of rotations. This means we will arrange the remaining 3 girls around her. The number of ways to arrange 4 girls in a circle is given by (n-1)!, where n is the number of girls.

\[ \text{Number of arrangements of girls} = (4-1)! = 3! = 6 \]

### Step 2: Identify the Gaps for Boys
Once the girls are seated, they create gaps where the boys can be placed. With 4 girls seated, there will be 4 gaps created between them (one gap before the first girl, one between each pair of girls, and one after the last girl).

### Step 3: Place the Boys in the Gaps
Since we need to ensure that no two boys sit next to each other, we can only place the boys in these gaps. We have 4 gaps and need to choose 3 of them to place the boys. The number of ways to choose 3 gaps from 4 is given by the combination formula:

\[ \text{Number of ways to choose gaps} = \binom{4}{3} = 4 \]

### Step 4: Arrange the Boys
After choosing the gaps, we can arrange the 3 boys in those selected gaps. The number of ways to arrange 3 boys is given by:

\[ \text{Number of arrangements of boys} = 3! = 6 \]

### Step 5: Calculate the Total Arrangements
Now, we can calculate the total number of arrangements by multiplying the number of arrangements of girls, the number of ways to choose gaps, and the number of arrangements of boys:

\[ \text{Total arrangements} = (\text{Arrangements of girls}) \times (\text{Ways to choose gaps}) \times (\text{Arrangements of boys}) \]

\[ \text{Total arrangements} = 6 \times 4 \times 6 = 144 \]

Thus, the total number of ways to arrange 3 boys and 4 girls around a circular table such that no two boys are together is **144**.