Question

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) $\dfrac{{^{52}{C_{26}}}}{{^{104}{C_{26}}}}$
B) $\dfrac{{2{ \times ^{52}}{C_{26}}}}{{^{104}{C_{26}}}}$
C) $\dfrac{{{2^{13}}{ \times ^{52}}{C_{26}}}}{{^{104}{C_{26}}}}$
D) \[\dfrac{{{2^{26}}{ \times ^{52}}{C_{26}}}}{{^{104}{C_{26}}}}\]

Answer 1

Hint: We have given some decks of playing cards which are well shuffled and some number of cards are randomly distributed to the players. In this problem our aim is to find the probability that the player gets all distinct cards. Let us find the required solution.

Complete step-by-step solution:
We have given the two decks of playing cards. Since one deck contains $52$ cards, which means two decks contains $52 \times 2 = 104$ cards. And these $104$ cards are well shuffled and from that $26$ cards are randomly distributed to the players.
That is, the number of ways of selecting $26$ cards out of $52$ cards can be expressed as $^{52}{C_{26}}$.
Since we know that in one deck there are $52$ cards and each distinct card is $2$ in number. So, therefore two decks will also contain only $52$ distinct cards two each.
So we can say that, probability that the player gets all the distinct cards is equal to $\dfrac{{2{ \times ^{52}}{C_{26}}}}{{^{104}{C_{26}}}}$.

Therefore, option (B) is the required solution.

Note: Answer for the question ‘’why are these $52$ cards in a deck of playing cards?’’ is the most common one that historians believe to be true is that $52$ cards represent $52$ weeks in a year. It can also be said that thirteen cards per suit could be representative of the thirteen lunar cycles in a year. Four suits may equal four seasons.
$^n{C_r}$ means if you give n different items and you have to choose r number of items from it, then $^n{C_r}$ gives the total number of ways possible. Here $^{52}{C_{26}}$ is the number of ways of selecting $26$ cards out of $52$ cards. And mathematically it can be written as $\dfrac{{52!}}{{26!\left( {52 - 26} \right)!}}$.