Answer:
This problem can be solved using the stars and bars technique.
Imagine we have 25 stars arranged in a row. We want to separate them into three groups to represent the values of x, y, and z. We can do this by placing two bars among the stars. For example:
||*******
The stars to the left of the first bar represent x, the stars between the two bars represent y, and the stars to the right of the second bar represent z. The number of stars to the left of the first bar represents the value of x, the number of stars between the two bars represents the value of y, and the number of stars to the right of the second bar represents the value of z.
Therefore, the problem is reduced to finding the number of ways we can place two bars among the 25 stars. There are 24 spaces between the stars where we can place the first bar, and 25 spaces where we can place the second bar. However, placing the second bar in the same space as the first bar would result in a group with zero stars, so there are only 24 spaces where we can place the second bar.
Therefore, the number of solutions is equal to the number of ways we can choose two out of the 24 spaces to place the bars. This can be calculated using the combination formula:
C(24, 2) = (24!)/(2!22!) = 276
Therefore, there are 276 solutions in nonnegative integers to the equation x+y+z=25.