Question

On Elm Street, there are 6 houses on one side of the street and 4 houses on the other. Each pair of houses on Elm Street is connected by exactly one telephone line. 

Column A	Column B
The total number of such lines that connect houses on opposite sides of Elm Street	12

Hint:

This type of problem is a classic application of the multiplication principle in combinatorics. Be careful to read the question to ensure you are connecting items between groups, not within the same group.

Answer 1

Step 1: Understanding the Concept:
This is a counting problem based on the fundamental counting principle. We need to find the total number of connections (lines) between two distinct groups of items (houses).
Step 2: Key Formula or Approach:
If there are \(m\) items in one group and \(n\) items in another group, and each item from the first group is connected to each item from the second group, the total number of connections is \(m \times n\).
Step 3: Detailed Explanation:
For Column A:
We have two groups of houses:
Group 1: 6 houses on one side of the street.
Group 2: 4 houses on the other side.
Each house from Group 1 must be connected to each house in Group 2.
Consider the first house in Group 1. It is connected to all 4 houses in Group 2. That's 4 lines.
The second house in Group 1 is also connected to all 4 houses in Group 2. That's another 4 lines.
This pattern continues for all 6 houses in Group 1.
So, the total number of lines is the number of houses in Group 1 multiplied by the number of houses in Group 2.
\[ \text{Total lines} = 6 \times 4 = 24 \] The value of Column A is 24.
For Column B:
The value is 12.
Step 4: Final Answer:
Comparing the two quantities:
Column A = 24
Column B = 12
Since \(24>12\), the quantity in Column A is greater.