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There are six people sitting at a circular table. Each person is either tall or short. Let a be the number of people sitting next to at least one tall person, and let b be the number of people sitting next to at least one short person. How many possible ordered pairs (a,b) are there? (For example, (a,b) = (6,0) if all six people are tall, since all six people are sitting next to a tall person, and zero people are sitting next to a short person.)

 Jan 20, 2020
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6 people

x be the number of people sitting next to at least one tall person

y be the number of people sitting next to at least one short person

 

(6,0) if all are short

(5,2)   if 1 tall person

(4,4)   if 2 tall people sitting together

(6,3)   if 2 tall people and they are separated by 1 seat

(6,4)   if 2 tall people are opposite each other

This pattern must be symmetrical

(0,6) if all are tall people

(2,5) if 1 short person

(4,4) if 2 short person and they sit together

(3,6) if 2 short people and they are separated by 1 seat

(4,6)   if the 2 short people are opposite each other

This one was harder.  But after drawing pics it seems to me that there are only 3 possibilities for 3 tall people and 3 short people.

(5,5)  The tall people all sit together

(5,5)  A pair of tall people, a pair of short people, a tall person, a short person

(6,6)  Alternating

 

so what do we have

(6,0), (0,6)

(5,2)(2,5)

(4,4)

(6,3)(3,6)

(6,4)(4,6)

(5,5)

(6,6)

 

11 pairs

 Jan 20, 2020

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