Competition problem — Combinatorics

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Combinatorics Difficulty 5.2 AIME, harder Find the answer

1. A deck of playing cards, excluding the big and small jokers, has a total of 52 cards. After shuffling, four people take turns to draw 13 cards each. The probability that the two red $\mathrm{A}$ s (i.e., the Heart A and the Diamond A) are in the same person's hand is $\qquad$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

$-1 . \frac{4}{17}$ .
Notice that, after the cards are shuffled, each person's cards are determined, i.e., the 52 cards are arranged in 52 positions. Let the four groups of card numbers be:
$\begin{array}{l} 1,5,9,13, \cdots, 49 ; 2,6,10,14, \cdots, 50 ; \\ 3,7,11,15, \cdots, 51 ; 4,8,12,16, \cdots, 52 . \end{array}$

Then the number of arrangements where the Ace of Hearts and the Ace of Diamonds are in the same group is $M=4 \mathrm{~A}_{13}^{2} \mathrm{~A}_{50}^{50}$ .
Thus, the required probability is $P=\frac{M}{52!}=\frac{4}{17}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.