Permutations and Combinations Question (1 Viewer)

Ambility

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In how many ways can the host and hostess and 4 visitors be arranged around a round table if the host and hostess sit next to each other?

My line of thinking:



The answer given in the answers section is 48. What did I do wrong?
 

InteGrand

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In how many ways can the host and hostess and 4 visitors be arranged around a round table if the host and hostess sit next to each other?

My line of thinking:



The answer given in the answers section is 48. What did I do wrong?
 

Ambility

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It still doesn't really make sense in my head as to why we divide by 5 instead of 6. When I imagine these problems I usually think of the amount of arrangements of people in a line and then think of how many ways these arrangements can be represented on a circle. Because each "straight line arrangement" can be rotated n times on a circle, and still have the same arrangement we must divide by n because there are n repetitions of the same arrangement.

I guess the fundamental thing I struggle with is, why do we consider groups instead of the people themselves?
 

leehuan

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The host and the hostess are arranged as one person, because they must always sit together no matter what.

The 4! occurs as a result of circular arrangement (where one person must be fixed), as opposed to a straight line arrangement which would have the 5! untouched. However, if the host and the hostess DIDN'T sit next to each other then
-> We don't arrange them as one person because they can sit anywhere
-> And there isn't the 2! for who sits on the left and the right.
 

leehuan

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(Gonna make a new post this time instead of edit an original:)

When two people sit together, we arrange them together because if we arrange them to be single, we have to somehow eliminate the cases where they are NOT sitting together. For convenience sake, by treating them as one person we don't have to insert a supposed subtraction sign.
 

Ambility

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(Gonna make a new post this time instead of edit an original:)

When two people sit together, we arrange them together because if we arrange them to be single, we have to somehow eliminate the cases where they are NOT sitting together. For convenience sake, by treating them as one person we don't have to insert a supposed subtraction sign.
I get that you treat them as one person and then swap their positions. That's why my original working was I don't get why it is a five on the denominator instead of six.
 

leehuan

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Like I said, the people are arranged together. If they're arranged together, you don't have a sixth person. The people you have are the 4 guests, and the one hosthostess

Total amount of people in your circular arrangement = 5
 
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