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Find the number of ways of choosing three people in a circular table with 12 people, so that no two people are next to each other. Use this image:

 Apr 10, 2024
 #1
avatar+128794 
+1

Not great at "counting" probs....here's my best attempt....corrections welcome !!!

 

1 3      5 6 7 8 9 10 11                7

2 4      6 7 8 9 10 11 12              7

3 5      7 8 9 10 11 12                 6

4 6      8 9 10 11 12 1                 6

5 7      9 10 11 12 1  2                6

6 8     10 11 12 1 2  3                 6

7 9     11 12 1  2  3  4                 6

8 10    12 1 2 3 4 5                     6

9 11     1 2 3 4 5 6                      6

10 12   2 3 4 5 6 7                      6

11  1    4 5 6 7 8                         5

12  2    5 6 7 8 9                         5

 

2(5) + 8(6) + 2(7)  =  72 ways

 

cool cool cool

 Apr 10, 2024
 #2
avatar+2440 
0

Tthe number of ways of choosing three people in a circular table with 12 people, so that no two people are next to each other.

 

Solution:

There are 12 ways to choose 3 persons who are adjacent (next to each other).

There are 12 ways to choose 2 persons who are adjacent.

There are 8 ways to choose 1 person who is not adjacent to the other two (2).

There are nCr(12,3) to choose a set of three persons from set of 12.

 

Subtract the persons who are adjacent from the total sets.

\(\text{So then ... } \dbinom {12}{3} - 12 - (12*8) = 112\\ \small \text{ There are 112 ways to select three persons from a set of 12 seated in a circle, such that no two persons are adjacent. }\\\)

 

GA

--. .-

GingerAle  Apr 11, 2024
edited by GingerAle  Apr 11, 2024

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