How many solutions are there to the equation x1 + x2 + x3 + x4 + x5 + x6 = 25 in which each xi is a non-negative integer and ... (a) There are no other restrictions. (b) xi ≥ 3 for i = 1, 2, 3, 4, 5, 6 (c) 3 ≤ x1 ≤ 10 (d) 3 ≤ x1 ≤ 10 and 2 ≤ x2 ≤ 7

The equation x1 + x2 + x3 + x4 + x5 + x6 = 25 in which each xi is a non-negative integer and ... Will have:

(a) 6 solutions where there are no other restrictions.

(b) 4 Solutions for xi ≥ 3 for i = 1, 2, 3, 4, 5, 6 (x3 + x4 + x5 + x6 = 25)

(c) 4 Solutions for 3 ≤ x1 ≤ 10

(x3 + x4 + x5 + x6 = 25)

(d) 5 Solutions for 2 ≤ x2 ≤ 7

Step-by-step explanation:

The equation have 6 unknown variables. This means it will have 6 solutions when no condition is imposed as in (a) above. The number of unknown variables in the equation will give the numbers of solutions

When the conditions are imposed, the numbers of unknown variables will give the number of solutions.

In (b) and (c), the equation will reduce to x3 + x4 + x5 + x6 = 25) and the unknown variables are 4, giving 4 solutions

In (d), the equation will reduce be 5 unknown variables (x2 + x3 + x4 + x5 + x6 = 25) and the solution is 5