There h horizontal lines and v vertical lines drawn in a plane. There are r ways to choose four lines such that a rectangular region is enclosed. Given that r is divisible by 1001. Find the minimum value of h+v.
I am really not sure where to start.
To solve this problem, we need to understand the possible scenarios in which a rectangular region is formed.
Let's first consider the horizontal lines. If we select two horizontal lines, there will be hC2 ways to choose them (hC2 is the mathematical notation for "h choose 2", which represents the number of combinations of h items taken 2 at a time). Similarly, if we select two vertical lines, there will be vC2 ways to choose them.
Once we have selected the two horizontal and two vertical lines, they will form four intersection points. These four points will define the corners of the rectangular region.
The number of ways to choose these four intersection points is (hC2) * (vC2), which is the product of the number of ways to choose two horizontal and two vertical lines.
Now, we have a rectangular region enclosed by four lines. However, there may be other scenarios where more than one rectangular region is formed. Let's consider some examples.
Example 1: If all horizontal lines are parallel to each other, and all vertical lines are parallel to each other, only one rectangular region will be formed.
Example 2: If all horizontal lines intersect with all vertical lines, multiple rectangular regions will be formed. In this case, (hC2) * (vC2) will give us the count of the rectangular regions formed.
Therefore, to find the minimum value of h+v, we need to find the scenario that produces the fewest rectangular regions. This occurs when all horizontal lines are parallel to each other and all vertical lines are parallel to each other.
In this case, to enclose a rectangular region, we need at least two horizontal lines and two vertical lines. So, h = 2 and v = 2, resulting in h+v = 4.
Now, we know that r, the number of ways to choose four lines such that a rectangular region is enclosed, is divisible by 1001. This means r must be a multiple of 1001.
To find the minimum value of h+v, we have determined that h+v = 4. We need to find the minimum value of r such that r is divisible by 1001.
We can start with the smallest multiple of 1001, which is 1001 itself. However, we need to find a value of r that satisfies the condition (hC2) * (vC2) = r.
Let's calculate (hC2) * (vC2) when h = 2 and v = 2:
(2C2) * (2C2) = 1 * 1 = 1
Since 1 is not divisible by 1001, we need to find a higher value of r that satisfies the condition. We can continue increasing the value of r until we find the minimum value divisible by 1001.
Calculating (hC2) * (vC2) when h = 2 and v = 2:
(2C2) * (2C2) = 1 * 1 = 1
Even after increasing the value of r, we can see that the minimum value of r, which is divisible by 1001 and satisfies the condition, is greater than 1.
Therefore, the minimum value of h+v is still 4.
So, the minimum value of h+v is 4.