How many distinct positive integer solutions (x, y, z) are there to the equation x + y + z = 10, where x, y, z >= 1?
Correct: B
This is a classic 'stars and bars' problem. We are looking for the number of positive integer solutions to x + y + z = 10.
Since x, y, z must be at least 1, we can use a transformation to convert this to a problem of non-negative integer solutions.
Let x' = x - 1, y' = y - 1, z' = z - 1.
Then x', y', z' >= 0.
Substitute these into the original equation:
(x' + 1) + (y' + 1) + (z' + 1) = 10
x' + y' + z' + 3 = 10
x' + y' + z' = 7.
Now we need to find the number of non-negative integer solutions to x' + y' + z' = 7.
Using the stars and bars formula, the number of solutions is given by C(n + k - 1, k - 1), where n is the sum (7 stars) and k is the number of variables (3 bars).
Number of solutions = C(7 + 3 - 1, 3 - 1) = C(9, 2).
C(9, 2) = 9! / (2! * (9-2)!) = 9! / (2! * 7!) = (9 * 8) / (2 * 1) = 72 / 2 = 36.
There are 36 distinct positive integer solutions.