Questions: A panel for a political forum is made up of 11 people from three parties, all seated in a row. The panel consists of 4 Labour Party members, 1 Socialist, and 6 Green Party members. In how many distinct orders can they be seated if two people of the same party are considered identical (not distinct)?

Solution Steps

We need to determine the number of distinct seating arrangements for a panel consisting of 11 members from three different parties: 4 Labour Party members, 1 Socialist member, and 6 Green Party members. Since members of the same party are considered identical, we will use the formula for permutations with repetition.

The formula for the number of distinct arrangements when there are indistinguishable objects is given by:

\[ \text{Distinct Orders} = \frac{n!}{n_1! \cdot n_2! \cdot n_3!} \]

where:

• \( n \) is the total number of items,
• \( n_1, n_2, n_3 \) are the counts of indistinguishable items.

In our case:

• \( n = 11 \) (total members),
• \( n_1 = 4 \) (Labour members),
• \( n_2 = 1 \) (Socialist member),
• \( n_3 = 6 \) (Green members).

We calculate the factorials:

• \( 11! = 39916800 \)
• \( 4! = 24 \)
• \( 1! = 1 \)
• \( 6! = 720 \)

Now we substitute these values into the formula:

\[ \text{Distinct Orders} = \frac{11!}{4! \cdot 1! \cdot 6!} = \frac{39916800}{24 \cdot 1 \cdot 720} \]

Calculating the denominator:

\[ 24 \cdot 1 \cdot 720 = 17280 \]

Now, we can compute the distinct orders:

\[ \text{Distinct Orders} = \frac{39916800}{17280} = 2310 \]

Final Answer