An ordinary deck of $52$ playing cards is randomly divided into $4$ groups of $13$ cards each. The probability that each group has exactly $1$ jack?
Step-by-Step Solution
Key Concept: The probability reduces to comparing the number of ways to distribute jacks (one per group) against all possible distributions of the 4 jacks among 4 groups.
We need to find the probability that when 52 cards are divided into 4 groups of 13 cards each, each group gets exactly one jack. The total number of ways to divide 52 cards into 4 groups of 13 is $\frac{52!}{(13!)^4}$. For favorable outcomes, we first assign the 4 jacks to the 4 groups (4! ways), then distribute the remaining 48 non-jack cards such that each group gets 12 more cards: $4! \times \frac{48!}{(12!)^4}$ ways. The probability is: $P = \frac{4! \times \frac{48!}{(12!)^4}}{\frac{52!}{(13!)^4}} = \frac{4! \times 48! \times (13!)^4}{52! \times (12!)^4} = \frac{24 \times (13)^4}{52 \times 51 \times 50 \times 49} \approx 0.1054$.
Correct Answer: 0.1054