<p>The number of integral solutions of \(x + y + z = 0\) with \(x \geq -5\), \(y \geq -5\), \(z \geq -5\) is</p>
Step-by-Step Solution
Key Concept: Transform the constrained problem into an unconstrained one using substitution x' = x+5, y' = y+5, z' = z+5, which converts the constraint inequalities into non-negativity conditions and shifts the equation accordingly.
<p><strong>Step 1:</strong> Apply substitution to remove lower bounds. Let x' = x+5, y' = y+5, z' = z+5.</p><p>Given constraints: x ≥ -5, y ≥ -5, z ≥ -5 become x' ≥ 0, y' ≥ 0, z' ≥ 0.</p><p><strong>Step 2:</strong> Transform the equation. Substitute x = x'-5, y = y'-5, z = z'-5 into x+y+z=0:</p><p>(x'-5) + (y'-5) + (z'-5) = 0</p><p>x' + y' + z' - 15 = 0</p><p>x' + y' + z' = 15</p><p><strong>Step 3:</strong> Count non-negative integer solutions. The number of non-negative integer solutions to x' + y' + z' = 15 is:</p><p>C(15+3-1, 3-1) = C(17, 2) = 17×16/2 = 136</p><p>∴ Answer: <strong>136</strong></p>
Correct Answer: B