In how many ways can two cards be drawn from a deck of 52 cards if the first card drawn is not replaced before the second card is drawn?
This is a very interesting question and problem. There are several different ways to approach the problem. With such a large array of ways to approach this question, a teacher is able to widen the horizon of his/her students pedagogically. Of course, the simplest and quickest way to solve the problem is to use the Counting Principle that says you multiply your number of choices for each card together. To answer this question, I am going to first use the Counting Principle in order to give myself a check for the other solutions I find. I suggest when using this problem in teaching a class that you do not introduce the Counting Principle first. The other methods discussed in this essay should be used to help introduce the Counting Principle.
Method 1: The Counting Principle
We are going to pick two cards at random from a deck of 52.
______*_____ These two blanks represent the two cards that are going to be drawn. Again, the Counting Principle tells us to find the total number of possibilities; we have to multiply the number of choices for each card together. Out of a deck of 52 cards, we have 52 choices (or 52 possible cards that could be drawn) for the first card. That means that 52 will go into our first blank. Because we don't put the first card that we pick back into the deck before picking the second card, we only have 52 - 1 = 51 choices for the second card. So 51 is the number that we fill into our second blank.
So, by using the Counting Principal, we have 52*51 = 2652 ways to draw two cards from a deck of 52 without replacing the first card before drawing the second card.
Method 2: Using a Spreadsheet
Often, students have trouble with thinking abstractly about this type of problem especially when it is being introduced to it for the first time. So, start with the basics. Talk about the make-up of a 52 deck of cards. One of the first things students will say when you ask what types of cards are in a deck of playing cards is "Two jokers." So, you have t o explain that the jokers have been taken out. Review the suits of the cards in the deck: H (hearts) , D (diamonds), C (clubs), S (spades) and discuss what value cards are in each suit: 2, 3, 4, …J, Q, K, A. It's also important to make sure that students understand that the pair 2 H 3 H is different from the pair 3 H 2 H. To help the students fully understand what they will be doing and to help get started, you can use a spreadsheet to find all the ways of your first card being a 2. Students can help or either come up with their own table of pairs of cards with the 2 of hearts as the first card. The table below shows all the possibilities for picking a 2 of hearts first, or does it?
2 H 3 H |
2 H 4 H |
2 H 5 H |
2 H 6 H |
2 H 7 H |
2 H 8 H |
2 H 9 H |
2 H 10 H |
2 H J H |
2 H Q H |
2 H K H |
2 H A H |
2 H 3D |
2 H 4D |
2 H 5 D |
2 H 6 D |
2 H 7 D |
2 H 8 D |
2 H 9 D |
2 H 10 D |
2 H J D |
2 H Q D |
2 H K D |
2 H A D |
2 H 3 S |
2 H 4 S |
2 H 5 S |
2 H 6 S |
2 H 7 S |
2 H 8 S |
2 H 9 S |
2 H 10 S |
2 H J S |
2 H Q S |
2 H K S |
2 H A S |
2 H 3 C |
2 H 4 C |
2 H 5 C |
2 H 6 C |
2 H 7 C |
2 H 8 C |
2 H 9 C |
2 H 10 C |
2 H J C |
2 H Q C |
2 H K C |
2 H A C |
One important fact to remember is that there are ways to pick two 2's as long as they are of different suits. Here are three more possible pairs that need to be included in the table: 2 H 2D , 2 H 2S , and 2 H 2 C .
So from the table above we have 4*12 = 48 ways to pick two cards with the first card being the 2 of hearts. We also have the three ways of picking two 2's with the first card being the 2 of hearts. That gives us a total of 48 + 3 = 51 possible ways to pick two cards from the deck with the first card being the two of hearts.
We now also know that there are also 51 ways to pick two cards with the first card being the 2 of diamonds, 51 ways of picking the 2 of clubs first, and 51 ways of picking the 2 of spades first. That means that there are 4*51 = 204 ways to choose two cards from the deck with the first card being a 2.
If we went through the same argument, there are 204 ways to pick two cards with the first card being a 3, 204 ways to have a 4 first, 204 ways to have a 5 first, 204 ways to pick a 6 first, etc. There are 204 ways to pick each of the following type of card first: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, and A. There are 13 different types of cards that could be picked first. So there are 13*204 = 2, 652 total ways to pick two cards from a deck of 52 without replacing the first card before drawing the second card. Notice that this is the same amount that we arrived at earlier when we used the Counting Principle.
Method 3: Another Spreadsheet
In this third method, we will set up another table to find the number of ways to draw two cards from a deck of 52 without replacement of the first card.
1^{ST}, 2nd |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
J |
Q |
K |
A |
2 |
2,2 |
2,3 |
2,4 |
2,5 |
2,6 |
2,7 |
2,8 |
2,9 |
2,10 |
2,J |
2,Q |
2,K |
2,A |
3 |
3,2 |
3,3 |
3,4 |
3,5 |
3,6 |
3,7 |
3,8 |
3,9 |
3,10 |
3,J |
3,Q |
3,K |
3,A |
4 |
…. |
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5 |
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6 |
…. |
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7 |
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8 |
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9 |
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10 |
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J |
…. |
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Q |
Q,2 |
Q,3 |
Q,4 |
Q,5 |
Q,6 |
Q,7 |
Q,8 |
Q,9 |
Q,10 |
Q,J |
Q,Q |
Q,K |
Q,A |
K |
K,2 |
K,3 |
K,4 |
K,5 |
K,6 |
K,7 |
K,8 |
K,9 |
K,10 |
K,J |
K,Q |
K,K |
K,A |
A |
A,2 |
A,3 |
A,4 |
A,5 |
A,6 |
A,7 |
A,8 |
A,9 |
A,10 |
A,J |
A,Q |
A,K |
A,A |
From the table above, we see that there are 13 rows and 13 columns of pairs of cards that can be drawn. So, there are 13*13 = 169 ways to draw 2 cards from a deck of 52.
Here are the different combinations of suits that are possible:
H H H D H C H S
D H D D D C D S As you can see, there are 16 different suit combinations.
C H C D C C C S
S H S D S C S S
So we have 169 * 16 = 2,704 ways to pick two cards out of a deck of 52. However, this number does not take into account that you can’t pick two 2s or two 3s of the same suit. (There is only one card of each suit.) We have to rule out the possibilities of picking two 2 H s, 2 D s, 2 C s, and 2 S s. There are 13 different value cards and 4 suits, giving us 13*4 =52 impossible pairs of duplicate cards that must be subtracted from the total number of ways to pick a pair of cards. So 2,704 — 52 = 2,652. There are 2,652 ways to pick two cards at random from a deck of 52 cards without replacing the first card before choosing the second card.
Again, I would suggest using methods 2 and 3 as ways of introducing the first method of using the Counting Principal. It often makes more sense for students to actually visualize what is being asked of them. It is also important for students to learn to set up tables as a method of solving some problems. However, many students will not need or want to use the table as a way of getting started. There are several other methods that can be used in order to answer the question posed. This problem could also be extended and some of the same methods could be used to solve those extensions.
Here are some possible extensions:
How many ways can three cards be selected from a deck of 52 without replacement of the first two drawn?
How many ways can four cards be picked from a deck of 52 without replacing each of the first three cards before drawing the next?
How many ways can n cards be drawn from a deck of 52 without replacing each of the first cards before picking the next?