WBJEE · Maths · Probability

Two decks of playing cards are well shuffled and 26 cards are randomly distributed to a player. Then, the probability that the player gets all distinct cards is

A \({ }^{52} C_{26} /{ }^{104} C_{26}\)
B \(2 \times{ }^{52} \mathrm{C}_{26} /{ }^{104} \mathrm{C}_{26}\)
C \(2^{3} \times{ }^{52} C_{26} /{ }^{104} C_{26}\)
D \(2^{26} \times{ }^{52} \mathrm{C}_{26} /{ }^{104} \mathrm{C}_{26}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(2^{26} \times{ }^{52} \mathrm{C}_{26} /{ }^{104} \mathrm{C}_{26}\)

Step-by-step Solution

Detailed explanation

Since, these are 52 distinct cards in decks and each distinct card is 2 in number. Therefore, 2 decks will also contain only 52 distinet cards two each. \(\therefore\) Probability that the player gets all distinct cards \[ =\frac{{ }^{52} C_{26} \times 2^{26}}{104_{C_{26}}} \]

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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