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Greg, Marcia, Peter, Jan, Bobby, and Cin...

Greg, Marcia, Peter, Jan, Bobby, and Cindy go to a movie and sit next to each other in six adjacent seats in the front row of the theater. If Marcia and Jan will not sit next to each other, in how many different arrangements can the six people sit?

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To solve the problem of how many different arrangements Greg, Marcia, Peter, Jan, Bobby, and Cindy can sit in a row such that Marcia and Jan do not sit next to each other, we can follow these steps: ### Step 1: Calculate the total arrangements without restrictions First, we need to find the total number of arrangements of the six people without any restrictions. The total arrangements can be calculated using the factorial of the number of people. \[ \text{Total arrangements} = 6! = 720 \] ### Step 2: Calculate the arrangements where Marcia and Jan sit together Next, we need to calculate the number of arrangements where Marcia and Jan are sitting next to each other. We can treat Marcia and Jan as a single unit or block. This means we now have five units to arrange: (MJ), Greg, Peter, Bobby, and Cindy. The number of arrangements of these five units is: \[ \text{Arrangements of 5 units} = 5! = 120 \] Since Marcia and Jan can be arranged within their block in two ways (MJ or JM), we multiply the arrangements of the five units by the arrangements within the block: \[ \text{Total arrangements with MJ together} = 5! \times 2 = 120 \times 2 = 240 \] ### Step 3: Calculate the arrangements where Marcia and Jan do not sit together To find the arrangements where Marcia and Jan do not sit next to each other, we subtract the number of arrangements where they are together from the total arrangements: \[ \text{Arrangements where MJ do not sit together} = \text{Total arrangements} - \text{Arrangements with MJ together} \] Substituting the values we calculated: \[ \text{Arrangements where MJ do not sit together} = 720 - 240 = 480 \] ### Final Answer Thus, the total number of arrangements where Marcia and Jan do not sit next to each other is: \[ \boxed{480} \]
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