Final answer:
The number of solutions to the equation x1 + x2 + x3 + x4 = 16 can be found using the inclusion-exclusion principle. When x1, x2, x3, and x4 are non-negative even integers, the number of solutions is C(11, 3). When x1, x2, x3, and x4 are integers with certain constraints, the number of solutions is C(21, 3). When x1, x2, x3, and x4 are non-negative integers with additional constraints, the number of solutions can be calculated using the inclusion-exclusion principle.
Step-by-step explanation:
To find the number of solutions to the equation x1+ x2+ x3+ x4 = 16, we can use the inclusion-exclusion principle.
a) If x1, x2, x3, and x4 are non-negative even integers, we can consider the equation as x1/2 + x2/2 + x3/2 + x4/2 = 8. This is equivalent to the equation y1 + y2 + y3 + y4 = 8, where y1, y2, y3, and y4 are non-negative integers. Using stars and bars, the number of solutions is C(8 + 4 - 1, 4 - 1) = C(11, 3).
b) If x1, x2, x3, and x4 are integers with x1 ≥ 0, x2 ≥ 2, x3 ≥ 0, and x4 ≥ 1, we can consider the equation as x1 + (x2 - 2) + x3 + (x4 - 1) = 16. This is equivalent to the equation x1 + x2 + x3 + x4 = 18. Using stars and bars, the number of solutions is C(18 + 4 - 1, 4 - 1) = C(21, 3).
c) If x1, x2, x3, and x4 are non-negative integers such that x1 ≤ 3, x2 ≤ 6, x3 ≤ 7, and x4 ≤ 5, we can consider the equation as x1 + x2 + x3 + x4 = 16. Using the inclusion-exclusion principle, we subtract the number of solutions where x1 > 3, x2 > 6, x3 > 7, or x4 > 5. The number of solutions where x1 > 3 is C(16 - 4 + 4 - 1, 4 - 1) = C(16, 3).
Therefore, the number of solutions to the equation x1 + x2 + x3 + x4 = 16 are:
a) C(11, 3)
b) C(21, 3)
c) C(16, 3) - C(13, 3) - C(10, 3) - C(9, 3) - C(12, 3) + C(9, 3) + C(6, 3) + C(5, 3)