For the following problems consider the equation X1 x2 x3

For the following problems consider the equation: X_1 + x_2 +x_3 = 15 How many integer solutions are there under the following conditions: x_i, e N for all i. x_1 Greaterthanorequalto 1, Greaterthanorequalto 3, x_3 Greaterthanorequalto 0 x_1 = 8. x_2 Greaterthanorequalto 3. x3 Greaterthanorequalto 0. 0 lessthanorequalto x_1 lessthanorequalto 6, x_2 Greaterthanorequalto 0. X_3 Greaterthanorequalto 0. (think of counting how many solutions have x_1 Greaterthanorequalto 7)

Solution

If the equation is in the for of x1+x2+...+xm = n, then the number of ways are n+m-1Cm-1 includeing 0.

so it becomes 17C2,excluding 0 it becomes 14C2 that is 91.

b) Assume y1 = x1-1, y2=x2-3, y3=x3, so equation becomes x1 + 1 +x2 +3 +x3=15

it becomes x1+x2+x3 = 11,

so it becomes 13C2 = 78.

c) Let x1=8, y1 = x2-3, y2=x3

so equation becomes 8+y1+3+y2 = 15 which becomes y1+y2 = 4, so it becomes 5c1 = 5.

d) let y1 = x1-7, so equation becomes y1+7 + y2+y3=15, so it becomes y1+y2+y3 = 8, so it can be chosen by 10 C 2 = 45.