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  • A deck of 52 playing cards is randomly shuffled. What is the probability that all the hearts cards are grouped together?Option: 1 <img alt="4 ! / 24

A deck of 52 playing cards is randomly shuffled. What is the probability that all the hearts cards are grouped together?

Option: 1

4 ! / 24 !


Option: 2

2 ! / 48 !


Option: 3

4 ! / 48 !


Option: 4

2 ! / 24 !


Answers (1)

best_answer

To calculate the probability that all the hearts cards are grouped together when a deck of 52 playing cards is randomly shuffled, we need to determine the favorable outcomes and the total number of possible outcomes.

Let's consider the hearts cards as a single group.

The number of ways to arrange the hearts cards among themselves is 4 !, since there are 4 hearts cards (Ace, 2, 3, and so on) that can be arranged among themselves.

The remaining cards (spades, diamonds, and clubs) can be arranged among themselves in 48 ! ways.

Therefore, the total number of possible outcomes (the denominator) is 48 !.

The favorable outcomes (the numerator) occur when the hearts cards are grouped together. Since the hearts cards can be arranged among themselves in 4 ! ways, and the remaining cards can be arranged in 48 ! ways, the number of favorable outcomes is 4 ! \times 48 !.

Therefore, the probability that all the hearts cards are grouped together is Simplifying further:

4 !=24

\begin{aligned} & 48 !=48 \times 47 \times 46 \times \ldots \times 2 \times 1 \\\\ & 4 ! / 48 ! \end{aligned}

Calculating this probability precisely requires a very large number of calculations, but it is a relatively small probability.

Therefore, the probability that all the hearts cards are grouped together when a deck of 52 playing cards is randomly shuffled is a relatively small value.

Posted by

Gautam harsolia

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