Question

Five-card stud is a poker game, in which a player is dealt 5 cards from an ordinary deck of 52 playing cards. How many distinct poker hands could be dealt?
A) ${}^{52}{C_5}$
B) $52!$
C) $13{!^4}$
D) None of these

Answer 1

Here, we will use the formula to calculate combination is ${}^n{C_r}$, where $n$ is the number of items, and $r$ represents the number of items being chosen. Then we will take $n = 52$ and $r = 5$ in the above way of combinations to find the distinct poker hands.

Complete step by step solution: We are given that the five-card stud is a poker game, in which a player is dealt 5 cards from an ordinary deck of 52 playing cards.
For this problem, it would be impractical for us to list all of the possible poker hands.
We know that the number of possible poker hands can be calculated using combinations.
We also know that the formula to calculate combination is ${}^n{C_r}$, where $n$ is the number of items, and $r$ represents the number of items being chosen.
Since we have 52 cards in the deck, so here $n = 52$.
Now we want to arrange them in unordered groups of 5, so here we have $r = 5$.
Substituting these values of $n$ and $r$ in the above formula of combinations, we get
$\Rightarrow {}^{52}{C_5}$
Therefore, there are ${}^{52}{C_5}$ distinct poker hands.

Hence, option A is correct.

Note:
In this problem, the order in which cards are dealt is not important, for example, if you dealt the ace, king, queen, jack, ten of spades, that is the same as being dealt the ten, jack, queen, king, ace of spades. Students use permutations ${}^n{P_r}$ instead of combinations, ${}^n{C_r}$, where $n$ is the number of items, and $r$ represents the number of items being chosen, which is wrong. The students can make an error by calculating the combinations but we just have to find a way of calculating. Here, students must take care while simplifying the conditions given in the question into the combinations.