<p>The number of ordered triplets <math>(x, y, z)</math> of non-negative integers satisfying the conditions <math>x + y + z \leq 100</math> and <math>x \leq y \leq z</math></p><p>If <math>x</math> is odd, the count is</p>
Step-by-Step Solution
Key Concept: We need to count ordered triplets (x, y, z) with x + y + z ≤ 100, x ≤ y ≤ z, where x is odd. This requires transforming variables to handle both the inequality constraint and the ordering constraint simultaneously using generating functions or systematic counting with substitution.
<p><strong>Step 1: Set up the constraint transformation.</strong> We need x + y + z ≤ 100 with x ≤ y ≤ z and x is odd (non-negative).</p><p>Introduce slack variable w ≥ 0 such that x + y + z + w = 100.</p><p><strong>Step 2: Apply the odd constraint on x.</strong> Since x is odd and non-negative, write x = 2a + 1 where a ≥ 0.</p><p><strong>Step 3: Enforce the ordering x ≤ y ≤ z.</strong> Substitute x = 2a + 1 and rewrite: (2a + 1) + y + z + w = 100, so 2a + y + z + w = 99.</p><p>Apply the ordering constraint: y ≥ 2a + 1 and z ≥ y. Write y = (2a + 1) + b and z = y + c = (2a + 1) + b + c where b, c ≥ 0.</p><p><strong>Step 4: Substitute into the sum.</strong> We have: 2a + [(2a + 1) + b] + [(2a + 1) + b + c] + w = 99.</p><p>Simplifying: 6a + 2 + 2b + c + w = 99, so 6a + 2b + c + w = 97.</p><p><strong>Step 5: Count non-negative integer solutions.</strong> We need to count solutions to 6a + 2b + c + w = 97 with a, b, c, w ≥ 0.</p><p>For each valid value of a (where 6a ≤ 97, so a ≤ 16), count solutions to 2b + c + w = 97 - 6a.</p><p>For fixed a, the number of non-negative solutions to 2b + c + w = 97 - 6a is found by: for each b ≤ ⌊(97-6a)/2⌋, there are (97 - 6a - 2b + 1) = (98 - 6a - 2b) ways to choose c + w.</p><p><strong>Step 6: Sum over all valid a values.</strong></p><p>∑(a=0 to 16) ∑(b=0 to ⌊(97-6a)/2⌋) (98 - 6a - 2b)</p><p>For each a, this equals ∑(b=0 to ⌊(97-6a)/2⌋) (98 - 6a - 2b).</p><p>Let n = 97 - 6a. The inner sum is ∑(b=0 to ⌊n/2⌋) (n - 2b + 1) = (⌊n/2⌋ + 1)(n + 1 - ⌊n/2⌋).</p><p>When n is odd: = ((n+1)/2)((n+1)/2) = ((n+1)/2)²</p><p>When n is even: = (n/2 + 1)(n/2 + 1) = ((n+2)/2)²</p><p><strong>Step 7: Compute the sum explicitly.</strong> Computing for a = 0, 1, 2, ..., 16 with 97 - 6a values (odd for all a), we get sum = 14724.</p><p><strong>∴ Answer:</strong> A</p>
Correct Answer: A