Permutations & Combinations
Linear Diophantine Equations
Grade 11
Question:
<p>Let N be the number of integral solution of the equation \(x + y + z + w = 15\) where \(x \geq 0\), \(y > 5\), \(z \geq 2\) and \(w \geq 1\). Find the unit digit of N.</p>
Step-by-Step Solution
Key Concept: Convert inequality constraints to equality constraints using substitution, then apply the stars and bars formula for distributing indistinguishable objects.
<p><strong>Step 1:</strong> Apply change of variables to convert to standard form.</p><p>Let $y = y' + 6$ (since $y > 5$ means $y \geq 6$), $z = z' + 2$, and $w = w' + 1$ where $y', z', w' \geq 0$.</p><p><strong>Step 2:</strong> Substitute into the equation:</p><p>$x + (y' + 6) + (z' + 2) + (w' + 1) = 15$</p><p>$x + y' + z' + w' = 6$</p><p><strong>Step 3:</strong> Find non-negative integer solutions using stars and bars:</p><p>$N = \binom{6+4-1}{4-1} = \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$</p><p><strong>Step 4:</strong> The unit digit of 84 is 4.</p>
Correct Answer: 4