Question

The face cards are removed from a full pack. Out of the remaining 40 cards, 4 cards are drawn at random. What is the probability that they belong to different suits?
A. $\dfrac{{1000}}{{9139}}$
B. $\dfrac{{1001}}{{9136}}$
C. $\dfrac{{100}}{{913}}$
D. $\dfrac{{1002}}{{9129}}$

Answer 1

There are 4 face cards in each suite. After all faces are removed, there are 10 cards left of each suit. We will find the number of ways in which 4 cards can be selected such that all four cards are of different suites by using the concept of combination. We will then find the probability using the formula, $\dfrac{{{\text{Number of possible outcomes}}}}{{{\text{Number of total outcomes}}}}$.

Complete step by step solution:
There are 40 cards left in the pack after 12 face cards have been removed from the full pack.
We have selected 4 cards from 40 cards.
If there are total $n$ objects and $r$ objects are to be selected, then the number of ways of doing it is by taking combination, $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Then, number of total ways of selecting 4 cards from 40 cards is $^{40}{C_4} = \dfrac{{40!}}{{4!36!}}$
On simplification we will get, $\dfrac{{40 \times 39 \times 38 \times 37 \times 36!}}{{4 \times 3 \times 2 \times 1 \times 36!}} = 91390$
Now, there are only 10 cards of each suite left.
We want to select one card from each suite.
Then the number of ways we can select 4 cards from the pack such that all the cards are of different suites.
$^{10}{C_1}{ \times ^{10}}{C_1}{ \times ^{10}}{C_1}{ \times ^{10}}{C_1} = {\left( {^{10}{C_1}} \right)^4}$
Also, $^n{C_1} = n$
Therefore the number of possible ways are ${10^4} = 10000$
The probability of selecting 4 cards from the pack such that all the cards are of different suite can be calculated using the formula, $\dfrac{{{\text{Number of possible outcomes}}}}{{{\text{Number of total outcomes}}}}$.
On substituting the values, we will get,
$\dfrac{{10000}}{{91390}} = \dfrac{{1000}}{{9139}}$

Hence, option A is correct.

Note:
Students must know that there are 4 suites in each full pack of cards, each having 13 cards, and out of these 4 are face cards and others are remaining. Here, the order of the selection does not matter. Hence, we have used the concept of combination.