x1 x2 x3 x4 11 how many solutions are there nonnegative

x₁ + x₂ + x₃ + x₄ = 11 ,how many solutions are there, non-negative integer, and positive integer (two different answers)

x₁ + x₂ + x₃ [less then or equal to] 14 ,how many solutions are there, non-negative integer, and positive integer (two different answers)

Solution

1) This is the same as the number of ways to choose 11 objects from 4
distinct objects with repetition and order does not matter,

non negative = 11+4-1 C_4-1 = 14C₃ = 364

positive

Solution.We set xi = yi +1 for i = 1; 2; 3; 4 and obtain y1 +y2 +y3 +y4 = 11-4 = 7.
So, there is one-to-one correspondence between the positive integer solutions of
x1 + x2 + x3+ x4 = 11 and the non-negative integer solutions of y1 + y2 + y3+ y4 = 7.
Hence, both equation have the same number of solutions. Since y1 + y2 + y3+ y4 = 7 has

7+4-1C_4-1 = 10C₃ = 120 solutions

2) x1+ x2+ x3 <= 14

turn this inequality to equality as x1+ x2+ x3+ x4 = 14

non negative = 14+4-1C₃ = 680

positive = 10+4-1C₃ = 286