<p>There are 12 balls numbered from 1 to 12. The number of ways in which they can be used to fill 8 places in a row so that the balls are with numbers in ascending or descending order is equal to</p>
Step-by-Step Solution
Key Concept: We need to count sequences where 8 balls from 12 are arranged in strictly ascending OR strictly descending order. Once we choose which 8 balls to use, there is exactly ONE way to arrange them in ascending order and exactly ONE way in descending order.
<p><strong>Step 1:</strong> To fill 8 places with balls in ascending order, we simply choose 8 balls from 12. Once chosen, there is exactly ONE way to arrange them in ascending order.</p><p>Number of ways = C(12,8) = 495</p><p><strong>Step 2:</strong> Similarly, to fill 8 places with balls in descending order, we choose 8 balls from 12, and there is exactly ONE way to arrange them in descending order.</p><p>Number of ways = C(12,8) = 495</p><p><strong>Step 3:</strong> Since these two cases are mutually exclusive (a sequence cannot be both strictly ascending AND strictly descending), we add them:</p><p>Total ways = C(12,8) + C(12,8) = 495 + 495 = 2 × C(12,8)</p><p>C(12,8) = C(12,4) = (12 × 11 × 10 × 9)/(4 × 3 × 2 × 1) = 11880/24 = 495</p><p>Total = 2 × 495 = <strong>990</strong></p><p>∴ Answer: D</p>
Correct Answer: D