How many ordered pairs of positive odd integers (x,y) satisfy x+y=1995?
None -- two odd numbers always have an even sum.
To find the number of ordered pairs of positive odd integers (x, y) that satisfy x + y = 1995, we need to consider the possible values for x and y.
Since x and y are positive odd integers, they can only take values from the set of positive odd integers. Let's represent the positive odd integers as 2k + 1, where k is a non-negative integer.
Substituting these representations into the equation x + y = 1995, we get (2k1 + 1) + (2k2 + 1) = 1995, where k1 and k2 are non-negative integers representing x and y, respectively.
Simplifying the equation, we get 2k1 + 2k2 + 2 = 1995, and dividing both sides by 2, we have k1 + k2 + 1 = 997.
Now we need to find the number of non-negative integer solutions for k1 and k2 that satisfy the equation k1 + k2 + 1 = 997.
Since k1 and k2 are independent variables, we can solve this equation using combinations. We have 997 options for the sum of k1 and k2, and we need to distribute this sum between k1 and k2.
Using the concept of stars and bars, we can say that k1 and k2 can be represented by two stars (**) and the sum of k1 and k2 is represented by 997 bars (||||||||||...||||||).
We need to align the stars and bars in such a way that both k1 and k2 are non-negative. To achieve this, we can align the first star with the first bar (|**||||||||||||...||||||) and there are 996 remaining bars.
Now we need to distribute the remaining 996 bars between the stars. Applying stars and bars combination formula, the number of ways to distribute bars (n - 1) between (r - 1) stars is given by (n - 1) C (r - 1), so the number of ordered pairs (k1, k2) is (996 + 1) C (2).
Calculating (997 C 2), we find that there are 495,006 ordered pairs of positive odd integers (x, y) that satisfy x + y = 1995.
To find the number of ordered pairs of positive odd integers that satisfy the equation x + y = 1995, we can use algebraic reasoning.
Let's consider the values that x can take. Since x and y are both positive odd integers, x must be an odd number. Therefore, x can be any odd number between 1 and 1994 (both inclusive). Note that if x exceeds 1994, y will be negative, violating the condition that y must be positive.
Now, if we know the value of x, we can calculate the value of y using the equation x + y = 1995. Rearranging this equation, we have y = 1995 - x.
To count the number of ordered pairs, we need to count the number of valid odd values of x. Since x can take any odd number between 1 and 1994, we can express this as the number of odd numbers between 1 and 1994 (inclusive).
To find the number of odd numbers between 1 and 1994, we use the formula:
(number of odd numbers) = (largest odd number - smallest odd number) / 2 + 1
In our case, the smallest odd number is 1, and the largest odd number is 1993. Plugging these values into the formula, we have:
(number of odd numbers) = (1993 - 1) / 2 + 1
= 1992 / 2 + 1
= 996 + 1
= 997
Therefore, there are 997 ordered pairs (x, y) of positive odd integers that satisfy x + y = 1995.