<p>Let 52 cards of a deck be arranged in a line. Number of ways in which cards appear in non-decreasing order of denomination is</p>
Step-by-Step Solution
Key Concept: Cards of same denomination are indistinguishable, so we need to count arrangements where all cards of denomination 1 come first, then denomination 2, etc. This becomes a problem of arranging identical objects in groups.
<p><strong>Step 1:</strong> Identify the structure. A standard deck has 13 denominations (A, 2, 3, ..., K) with 4 cards each (4 suits).</p><p><strong>Step 2:</strong> For non-decreasing order of denomination, all 4 cards of denomination 1 must appear first, then all 4 of denomination 2, etc. Since cards of the same denomination are identical in terms of denomination ordering, we don't distinguish between them.</p><p><strong>Step 3:</strong> The total number of ways to arrange 52 cards in non-decreasing order of denomination is simply <strong>1</strong> — there is only one way: place the 4 cards of denomination 1, then 4 of denomination 2, and so on through denomination 13.</p><p><strong>Step 4:</strong> However, if the question asks about distinguishing suits as well (which is typical for such problems), we need the multinomial coefficient:</p><p>$$\binom{52}{4,4,4,...,4} = \frac{52!}{(4!)^{13}}$$</p><p>This counts arrangements where we have 4 indistinguishable positions for each denomination group.</p><p>∴ Answer: D</p>
Correct Answer: D