Eight people are put into pairs and enter a room with a large table with four pairs of chairs (2 chairs and a lot of space then 2 more chairs and a lot of space and so on) Though they could sit anywhere they had to sit with there assigned partner. In How Many ways could the eight people have sat around the table ensuring that each pre-assigned pair was sitting at a pre-set pair of the chairs
The first assigned pair has four pairs of chairs to choose from. The second has three and the third has two. In each case, the pair can do a left-right switch. That makes 4x3x2x2 = 48 possibilities.
To determine the number of ways the eight people can sit around the table while ensuring that each pre-assigned pair sits at a pre-set pair of chairs, we can break down the problem into several steps:
Step 1: Assign the first pair of people to a pair of chairs. There are 4 pairs of chairs, so we have 4 options for the first pair.
Step 2: Assign the second pair of people to one of the remaining pairs of chairs. Since one pair has already been assigned, there are only 3 pairs of chairs left to choose from. Therefore, we have 3 options for the second pair.
Step 3: Continue this process for the remaining pairs of people. At each step, the number of options decreases by 1.
So, the total number of ways the eight people can sit around the table is calculated by multiplying the number of options at each step:
4 options for the first pair * 3 options for the second pair * 2 options for the third pair * 1 option for the fourth pair = 4 * 3 * 2 * 1 = 24.
Thus, the eight people could sit around the table in 24 different ways while ensuring that each pre-assigned pair sits at a pre-set pair of chairs.