<p>Let 52 cards of a deck be arranged in a line. Number of ways in which spades appear in increasing order of denomination is</p>
<p>\(\dfrac{52!}{13!}\)</p>
<p>\({}^{52}C_{13} \times 2^{39}\)</p>
<p>\({}^{52}C_{13} \times 13!\)</p>
<p>\(\dfrac{52!}{13! \cdot 13!}\)</p>
Step-by-Step Solution
Key Concept: Once we select which 13 positions out of 52 will hold the spades, there is exactly ONE way to arrange them (in increasing order). The problem reduces to choosing 13 positions from 52 for the spades, and the remaining 39 cards can be arranged freely.
<p><strong>Step 1:</strong> Identify the constraint: Among 52 cards arranged in a line, the 13 spades must appear in increasing order of denomination (A, 2, 3, ..., K).</p><p><strong>Step 2:</strong> Key insight: Once we select which 13 positions (out of 52) will contain the spades, there is exactly ONE way to fill them (increasing order is forced). We cannot choose any other arrangement of spades in those positions.</p><p><strong>Step 3:</strong> Choose 13 positions from 52 for spades: C(52,13) ways</p><p><strong>Step 4:</strong> Arrange the remaining 39 cards (hearts, diamonds, clubs) in the remaining 39 positions: 39! ways</p><p><strong>Step 5:</strong> Total arrangements = C(52,13) × 39! = $\frac{52!}{13! \times 39!} \times 39! = \frac{52!}{13!}$</p><p><strong>Alternative form:</strong> This equals P(52,13) or (52 × 51 × 50 × ... × 40)</p><p>∴ Answer: A</p>
Correct Answer: A