Final answer:
To find the number of solutions to the equation x1 x2 x3 x4 x5 x6 = 25, we can use the concept of partitions. By visualizing the problem as distributing stars among bins, we can apply the stars and bars method to calculate the number of solutions. In this case, there are a total of 230,230 solutions to the equation with no other restrictions.
Step-by-step explanation:
The equation x1 x2 x3 x4 x5 x6 = 25 represents a product of six variables with a result of 25. To determine the number of solutions, we need to find the number of ways to distribute the factor of 25 among the six variables. Since each xi can be a non-negative integer, this is equivalent to finding the number of partitions of 25 into six parts.
We can solve this using a combinatorial approach. By applying the stars and bars method, we can visualize this problem as distributing 25 stars (representing the 25) among six bins (representing the six variables). The number of ways to do this is given by the formula C(25+6-1, 6-1), where C(n, r) represents the number of combinations of n objects taken r at a time.
Calculating C(25+6-1, 6-1) gives us a total of 230,230 solutions to the equation x1 x2 x3 x4 x5 x6 = 25 with no other restrictions.