要从5名女性和6名男性中选出一个由六人组成的委员会。其中三名女性是姐妹。在以下情况下，求可以组成委员会的方法数： 所选委员会由两名特定的姐妹、另外3名男性和另外1名女性组成。他们坐在一个专供六人使用的圆桌旁。在以下情况下，求可能的排列数：

1. Identify the number of women needed for the committee

To form a committee of $6$ people with exactly $2$ men, the number of women required is $6 - 2 = 4$.

2. Calculate the total number of ways to form the committee

Number of ways to choose $2$ men out of $6$ and $4$ women out of $5$:
$^6C_2 \times ^5C_4 = 15 \times 5 = 75$

1. Calculate the total number of ways to form the committee without restrictions

Total number of people = $5 \text{ women} + 6 \text{ men} = 11$
Total unrestricted ways to choose 6 people = $\binom{11}{6} = 462$

2. Calculate the number of ways to form the committee without including any of the sisters

Number of people who are not sisters = $11 - 3 = 8$
Ways to choose a committee of 6 entirely from these 8 people = $\binom{8}{6} = 28$

3. Subtract the ways without sisters from the total ways to find the number of ways with at least one sister

\text{Ways with at least one sister} = 462 - 28 = 434

1. Find the number of ways to arrange the two sisters and the man between them into a single block.

We choose $1$ man out of the $3$ available men on the committee to sit between the sisters: $\binom{3}{1} = 3$ ways.

The two sisters can swap places on either side of the chosen man in $2! = 2$ ways.

Treating the two sisters and this chosen man as a single block, there are $3 \times 2 = 6$ ways to form this block.

2. Calculate the total number of circular arrangements for all entities.

We now arrange this block along with the remaining $3$ people ($2$ men and $1$ woman) around the round table.

Total number of entities to arrange $= 1 \text{ (block)} + 3 \text{ (individuals)} = 4$ entities.

The number of ways to arrange $4$ entities in a circle is $(4 - 1)! = 3! = 6$ ways.

Total possible arrangements $= 6 \times 6 = 36$.

1. Place the two sisters at the round table.

In a circular arrangement, fixing the position of one person removes the rotational symmetry, leaving $1$ way to seat the first sister.
Since the two sisters must sit directly opposite each other, the second sister has only $1$ available seat (directly across from the first).
Number of ways to seat the sisters = $1 \times 1 = 1$.

2. Arrange the remaining committee members in the remaining seats.

The remaining $4$ members (the $3$ men and $1$ other woman) can be arranged in the remaining $4$ distinct seats in $4!$ ways.
Total number of possible arrangements = $1 \times 4! = 24$.