<p>Let <i>n</i> be a four-digit integer in which all the digits are different. If <i>x</i> is number of odd integers and <i>y</i> is number of even integers, then</p>
Step-by-Step Solution
Key Concept: For a four-digit number with distinct digits, the parity (odd/even) is determined solely by the units digit. Odd numbers need odd units digits (1,3,5,7,9) while even numbers need even units digits (0,2,4,6,8), and the remaining three positions have equal flexibility in both cases.
<p><strong>Step 1:</strong> For odd four-digit integers with distinct digits, units digit must be from {1,3,5,7,9} (5 choices). First digit: 8 choices (1-9 except the units digit). Second digit: 8 choices (0-9 except first and units). Third digit: 7 choices. So x = 5 × 8 × 8 × 7 = 2240.</p><p><strong>Step 2:</strong> For even four-digit integers, units digit from {0,2,4,6,8}. <br/>• Case 1 (units digit = 0): First digit has 9 choices, second has 8, third has 7. Subtotal = 9 × 8 × 7 = 504.<br/>• Case 2 (units digit ∈ {2,4,6,8}): 4 choices for units. First digit: 8 choices (1-9 except units digit). Second digit: 8 choices. Third digit: 7 choices. Subtotal = 4 × 8 × 8 × 7 = 1792.<br/>So y = 504 + 1792 = 2296.</p><p><strong>Step 3:</strong> Comparing x and y: x = 2240 < y = 2296, so x < y.</p><p>∴ Answer: B</p>
Correct Answer: B