Question

4. Determine the number of integer solutions of x1 + x2 + x3 + x4 = 17, where

a. xi ≥ 0, 1 ≤ i ≤ 4

b. x1, x2 ≥ 3 and x3, x4 ≥ 1

c. xi ≥ -2, 1 ≤ i ≤ 4

d. x1 , x2 , x3 > 0 and 0 < x4 ≤ 10

Answer 1

This is a case of a multinomial distribution.

a) Total number of solutions here such that each number is greater than 0 is computed using the multinomial distribution formula as:

b) Let y₁ = x₁ - 2 and y₂ = x₂ - 2.

Therefore, if x₁, x₂ >= 3, then y_1, y₂ >= 3 - 2 = 1

Therefore, we have here:

y₁ + y₂ + x₃ + x₄ = 17 - 2 - 2

y₁ + y₂ + x₃ + x₄ = 13

Now the number of solutions is computed here using the same formula as:

c) We are given here that:

x_i >= -2 here for all i, lets keep: y_i = x_i + 3 for all i.

Then, we have here:

y₁ + y₂ + y₃ + y₄ = 17 + 3*4 = 29

Therefore, the number of solutions here is computed as:

d) Here, as each of x_i for all i is greater than 0, therefore it has to be greater than equal to 1.

Also, we are given that all x₄ <= 10

The number of solutions here is computed as:

Let x₄ > 10

Then keeping y₄ = x₄ - 10, we have: y₄ > 0

Therefore,

x₁ + x₂ + x₃ + y₄ = 17 - 10 = 7

Number of solutions of above equation is computed as:

And total number of solutions:

Therefore total number of allowable solutions is computed as:

= 560 - 20 = 540

Therefore 540 is the required number of solutions here.