Let "n" be a positive integer. How many points (x,y) in the coordinate plane are there such that x and y are integers and 0<y<x<n?
(n - 1)(n - 2)
Scott, does mean that there is only one point for this problem?
Is it (2,1)
Can you explain why you set it up as 2 binomials? Any further explanation would be greatly appreciated..
Thank you so much.
To find the number of points (x, y) in the coordinate plane such that x and y are integers and 0 < y < x < n, we can break down the problem into smaller steps.
Step 1: Understanding the constraints
- We are looking for points (x, y) where both x and y are integers.
- The y-coordinate should be greater than 0 and less than the x-coordinate.
- The x-coordinate should be less than n, where n is a positive integer.
Step 2: Analyzing the constraints
Let's start by considering the possible values for y when x = 1.
- Since 0 < y < x, the only possible value for y is 1.
For x = 2, there are two possible values for y (0 < y < 2):
- y = 1 and y = 2.
For x = 3, there are three possible values for y (0 < y < 3):
- y = 1, y = 2, and y = 3.
Following this pattern, we can observe that for each x greater than 1, the number of possible values for y increases by one.
Step 3: Counting the points
Let's consider an example with n = 5:
- For x = 1, the only possible point is (1, 1).
- For x = 2, the possible points are (2, 1) and (2, 2).
- For x = 3, the possible points are (3, 1), (3, 2), and (3, 3).
- For x = 4, the possible points are (4, 1), (4, 2), (4, 3), and (4, 4).
Therefore, there are 1 + 2 + 3 + 4 = 10 points in total.
Step 4: Generalizing the pattern
To find the total number of points for any positive integer n, we need to calculate the sum of all positive integers from 1 to n-1.
This can be done using the formula for the sum of an arithmetic series:
Sum = (n-1) * ((n-1) + 1) / 2
So, the number of points (x, y) satisfying the given conditions is:
Sum = (n-1) * (n-1+1) / 2 = (n-1) * n / 2 = n*(n-1) / 2
Therefore, the number of points is n*(n-1) / 2.