Twelve couples, who live in the Happy
Retirement Villa, decide to install intercom systems between each of their 12 suites. How many connecting lines are necessary to permit direct conversation between any suites?
(a) 2 couples
(b) 3 couples
(c) 4 couples
(d) 5 couples
Thank you
I have no idea
Your choices do not match the question. The answer should be in units of connecting lines, not in the number of couples.
I suspect someone copied the problem incorrectly.
To determine how many connecting lines are necessary to permit direct conversation between any suites, you need to consider the number of possible pairs of suites within the twelve couples in the Happy Retirement Villa.
Since there are 12 couples, there are 12 suites in total.
To calculate the number of possible pairs, you can use the combination formula:
C(n, r) = n! / (r! * (n - r)!)
Where n is the total number of suites, and r is the number of suites in each pair.
For option (a), where there are 2 couples:
C(4, 2) = 4! / (2! * (4 - 2)!)
= 4! / (2! * 2!)
= (4 * 3 * 2 * 1) / (2 * 1 * 2 * 1)
= 24 / 4
= 6
For option (b), where there are 3 couples:
C(6, 2) = 6! / (2! * (6 - 2)!)
= 6! / (2! * 4!)
= (6 * 5 * 4 * 3 * 2 * 1) / (2 * 1 * 4 * 3 * 2 * 1)
= 720 / 48
= 15
For option (c), where there are 4 couples:
C(8, 2) = 8! / (2! * (8 - 2)!)
= 8! / (2! * 6!)
= (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (2 * 1 * 6 * 5 * 4 * 3 * 2 * 1)
= 40320 / 2880
= 14
For option (d), where there are 5 couples:
C(10, 2) = 10! / (2! * (10 - 2)!)
= 10! / (2! * 8!)
= (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (2 * 1 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
= 3628800 / 80640
= 45
Therefore, the correct answer is (d) 5 couples, which would require 45 connecting lines.