Final answer:
The number of non-negative integer solutions to the equation x+y+z+t = 15 is 816, calculated using the stars and bars method, which is a combinatorial technique.
Step-by-step explanation:
The question asks for the number of non-negative integer solutions to the equation x+y+z+t = 15. This problem is an example of a combinatorial problem involving partitioning a number into a set of summands. To count the number of solutions, we can use the stars and bars method (a combinatorial technique).
The stars and bars method involves imagining the sum 15 as 15 'stars', and we need to place 3 'bars' to divide the stars into 4 groups representing x, y, z, and t. The total number of ways to arrange these stars and bars is given by choosing 3 spots out of 18 (15 stars + 3 bars), because we can convert any arrangement of 15 stars and 3 bars into a solution, and vice versa.
Therefore, the number of non-negative integer solutions is the number of combinations of choosing 3 spots from 18, which is calculated using the binomial coefficient C(18,3). The formula for the binomial coefficient is C(n,k) = n! / (k!(n-k)!), and hence the number of solutions is C(18,3) = 18! / (3!15!) = 816.