Question

From a deck of 52 cards, how many different four-card hands could be dealt which include one card from each suit?

Answer 1

Hint: We know that a deck of 52 cards contains four suits. They are hearts, spades, diamonds and clubs. We have to select one card from each of these suits to make a four-card hand. Each suit has 13 cards, therefore we have to select one from these 13 cards for each hand.

Complete step by step solution:
To make one four card hand with all different cards, we need to choose a card from each suit that is one from hearts, one from spades, one from diamonds and one from clubs.
Thus, we have to select one card from 13 cards of each suit first.
The number of ways in which this can be done is : $\left( {\begin{array}{*{20}{c}}
  {13} \\
  1
\end{array}} \right) = 13$.
As we need to make the hands in which all for cards are different, the number of ways for this can be determined by multiplying the previous calculation four times which is:
$\left( {\begin{array}{*{20}{c}}
  {13} \\
  1
\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}
  {13} \\
  1
\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}
  {13} \\
  1
\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}
  {13} \\
  1
\end{array}} \right) = {13^4} = 28561$
Thus, our final answer is: 28561 different four-card hands could be dealt which include one card from each suit.
So, the correct answer is “28561”.

Note: This type of question can also be asked where we have to make hands in which all the cards are of the same suit.
In this, we have to select one of the four suits which can be done in $\left( {\begin{array}{*{20}{c}}
  4 \\
  1
\end{array}} \right) = 4$ ways.
Now, we have to select four cards from the already selected suit which can be done in $4 \times \left( {\begin{array}{*{20}{c}}
  {13} \\
  4
\end{array}} \right) = 4 \times \dfrac{{13 \times 12 \times 10 \times 9}}{{4 \times 3 \times 2 \times 1}} = 2860$ ways.