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?
Algebra - Scott yesterday at 9:05pm
(n - 1)(n - 2)
Algebra - Laurie yesterday at 9:55pm
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 calculate the number of points (x,y) that satisfy the given conditions, where 0 < y < x < n, we need to consider the range of possible values for x and y.
1. Let's start by considering the possible values for x. Since 0 < y < x < n, we know that x can take values from 1 to n-1. So, the range of possible values for x is {1, 2, 3, ..., n-1}.
2. Now, let's consider the possible values for y. For each value of x, the range of possible values for y is from 1 to x-1 (since y < x). So, for x = 2, the possible values for y are {1}, for x = 3, the possible values for y are {1, 2}, and so on.
3. To count the total number of points (x,y), we need to sum up the number of possible values for y for each value of x. For x = 2, there is 1 possible value for y. For x = 3, there are 2 possible values for y. And so on, for x = 4, there are 3 possible values for y, and for x = 5, there are 4 possible values for y.
4. If we sum up the number of possible values for y for each value of x, we get 1 + 2 + 3 + ... + (n-2). This is the sum of the first (n-2) positive integers.
5. The sum of the first (n-2) positive integers can be calculated using the formula for the sum of an arithmetic series: S = (n-2)(n-1)/2.
Therefore, the number of points (x,y) in the coordinate plane such that 0 < y < x < n is given by (n-2)(n-1)/2, as mentioned by Scott.
So, for the given problem, the number of points is (n-2)(n-1)/2. It does not represent a specific point like (2,1).