Question

From a pack of 52 playing cards, two cards are drawn together at random. Calculate the probability of both the cards being the Kings.
A. $\dfrac{1}{15}$
B. $\dfrac{25}{57}$
C. $\dfrac{35}{256}$
D. None of the above

Answer 1

Hint: First we will write down what a deck of cards contain and how many kings it contains and then we will find the number of total cases by applying $^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$ for drawing 2 cards out of 52 cards then we will again apply this to find out the number of conditional cases. Finally, we will put it in the probability formula that is $\text{Probability}=\dfrac{\text{Number of conditional cases}}{\text{Total number of cases}}$ and get the required answer.

Complete step-by-step solution
We have a total number of cards $=52$ and in each suit, the number of cards is $=13$ and there are a total of $4$ type of suits that are hearts, diamond, club, and spades and in each suit, we will have one king so the total number of the cards being the Kings is 4.
First, of all let’s see how can we draw two cards out of 52 cards, for this we will apply the combination formula: \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\], where n is the total number of objects and r is the selected number of objects, therefore:
The number of ways in which we can draw 2 cards from 52 cards is:
 ${}^{52}{{C}_{2}}=\dfrac{52!}{2!\left( 52-2 \right)!}=\dfrac{50!\times 51\times 52}{2\times 50!}=1326$ ways.
Now we know that the probability of an event happening is as following:
$\text{Probability}=\dfrac{\text{Number of conditional cases}}{\text{Total number of cases}}$
Now let’s find out the total number of conditional cases, we are given the conditional cases is to select two cards and both of them should be Kings, therefore we will select 2 cards out of 4 king cards: ${}^{4}{{C}_{2}}=\dfrac{4!}{2!\left( 4-2 \right)!}=6$ ways.
Therefore, the probability that the two cards are drawn from a pack of 52 cards are kings is:
$\dfrac{6}{1326}=\dfrac{1}{221}$
Hence, the correct is option D: None of the above.

Note: Students can mistakes in applying $^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$ as the number are bigger so the calculation might get messy so one needs to be careful there and also student may apply the standard formula for probability and can write it as $\dfrac{4}{52}$ for 4 kings out of 52 cards, these mistakes must be avoided.