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 card is

• A. $\frac{{}^{52}{\mathrm{C}}_{26}}{{}^{104}\mathrm{C}_{26}}$
• B. $\frac{2\times {}^{52}{\mathrm{C}}_{26}}{{}^{104}\mathrm{C}_{26}}$
• C. $\frac{{\left(2\right)}^{13}\times {}^{52}{\mathrm{C}}_{26}}{{}^{104}\mathrm{C}_{26}}$
• D. $\frac{{\left(2\right)}^{26}\times {}^{52}{\mathrm{C}}_{26}}{{}^{104}\mathrm{C}_{26}}$

Answer 1

The correct option is D

$\frac{{\left(2\right)}^{26}\times {}^{52}{\mathrm{C}}_{26}}{{}^{104}\mathrm{C}_{26}}$

Find the required probability:

There are $52$ cards in a deck of cards.

Hence, there are $104$ cards in a two decks.

$26$ cards can be selected from $104$ cards in ${}^{104}\mathrm{C}_{26}$ ways.

Hence, total number of favourable outcomes $={}^{104}\mathrm{C}_{26}$

There are only $52$ distinct cards in two decks.

So we select a card from either deck and we have $26$ selections to make.

Hence, the number of ways of choosing $26$ distinct cards is given as ${}^{52}{\mathrm{C}}_{26}\times {\left({}^2\mathrm{C}_1\right)}^{26}$

Hence the probability of getting $26$ distinct cards $=\frac{\mathrm{number}\mathrm{of}\mathrm{favorable}\mathrm{outcomes}}{\mathrm{number}\mathrm{of}\mathrm{total}\mathrm{outcomes}}$

$=\frac{{}^{52}{\mathrm{C}}_{26}\times {\left({}^2\mathrm{C}_1\right)}^{26}}{{}^{104}\mathrm{C}_{26}}$

$=\frac{{}^{52}{\mathrm{C}}_{26}\times {\left(2\right)}^{26}}{{}^{104}\mathrm{C}_{26}}$

Therefore, the required probability is $\frac{{\left(2\right)}^{26}\times {}^{52}{\mathrm{C}}_{26}}{{}^{104}\mathrm{C}_{26}}$.

Hence, option (D) is the correct answer.