A committee consisting of six people is to be selected from five women and six men. Three of the women are sisters. Find the number of ways the committee can be formed if The chosen committee consists of two particular sisters together with 3 other men and one other woman. They are seated at a round table meant for six persons. Find the number of possible arrangements if

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 = $1 \times 4! = 24$.