Question

A standard deck of cards has 52 cards. If you draw one card at a time from the deck, without replacement, how many different ways can you draw all of the cards? Remember that "without replacement" means that the cards are not returned to the deck after they are chosen.Write your answer in factorial notation.

Answer 1

When drawing the first card, there are 52 possible outcomes, that is, 52 different cards.

Then, for the second card, there is one less, so we have 51 possible outcomes.

This way, each card drawn will reduce the number of outcomes by 1.

The total number of different ways is given by the product of each number of outcomes from each card, so we have:

$N=52\cdot51\operatorname{\cdot}50\operatorname{\cdot}...\operatorname{\cdot}3\operatorname{\cdot}2\operatorname{\cdot}1$

Using the factorial notation, we can rewrite N as follows:

$\begin{gathered} x!=x\operatorname{\cdot}(x-1)\operatorname{\cdot}(x-2)\operatorname{\cdot}...\operatorname{\cdot}2\operatorname{\cdot}1\\ \\ \\ \\ N=52! \end{gathered}$

Therefore the number of different ways is the factorial of 52.