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
To determine the number of connecting lines necessary to permit direct conversation between any suites, we need to consider that each suite needs to be connected to every other suite, excluding itself.
We have 12 suites, and we can choose 2 suites at a time to connect. This is referred to as a combination.
The formula for calculating combinations is given by:
C(n, r) = n! / (r! * (n-r)!),
where n is the total number of items and r is the number of items chosen at a time.
In this case, we want to choose 2 suites at a time out of 12, so the calculation would be:
C(12, 2) = 12! / (2! * (12-2)!)
Now let's calculate this value:
C(12, 2) = (12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((2 * 1) * (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1))
Simplifying the expression:
C(12, 2) = (12 * 11) / (2 * 1)
= 132 / 2
= 66
Therefore, to permit direct conversation between any suites in the Happy Retirement Villa, we would need 66 connecting lines.
Based on the given options:
(a) 2 couples - Does not match with the calculated answer.
(b) 3 couples - Does not match with the calculated answer.
(c) 4 couples - Does not match with the calculated answer.
(d) 5 couples - Does not match with the calculated answer.
None of the given answer options match with the calculated answer of 66. It seems there might be an error in the options or in the given information.