Counting Techniques Question: Drawing 5 specific cards out of a deck

In summary, for the given scenario of drawing 5 cards from a deck of 52 without replacement, the number of combinations with exactly one Jack, one Queen, and one King is 49,920. This can be calculated by choosing 1 card from each of the 4 Jacks, Queens, and Kings (4C1 x 4C1 x 4C1 = 64) and then choosing 2 more cards from the remaining 40 (40C2 = 780). This results in a total of 64 x 780 = 49,920 combinations. The previous attempt of 99,840 was incorrect because it did not take into account that the remaining two cards can be chosen in any order,
  • #1
student74
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Homework Statement



We draw 5 cards out of a normal deck of 52 cards. No replacements.
How many combinations have exactly one Jack, one Queen, one King?

Homework Equations





The Attempt at a Solution



The solution = 49,920

My attempts:

I keep getting 99,840.

So 4C1 Jack, 4C1 Queen, 4C1 King, then for remaining two cards 40 possible, then 39 possible.

I go 4C1 x 4C1 x 4C1 x 40 x 39 = 99,840

Are there combinations that are repeated somewhere then? :S
 
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  • #2
student74 said:

Homework Statement



We draw 5 cards out of a normal deck of 52 cards. No replacements.
How many combinations have exactly one Jack, one Queen, one King?

Homework Equations





The Attempt at a Solution



The solution = 49,920

My attempts:

I keep getting 99,840.

So 4C1 Jack, 4C1 Queen, 4C1 King, then for remaining two cards 40 possible, then 39 possible.

I go 4C1 x 4C1 x 4C1 x 40 x 39 = 99,840

Are there combinations that are repeated somewhere then? :S

After choosing 1 Jack, 1 Queen and 1 King you need to choose two more cards from 40. The number of ways of doing that =?

RGV
 
  • #3
oh! thanks Ray! Why doesn't it work going 40 x 39 instead of 40C2? Does 40 x 39 not work because that takes into account order as being relevant?
 

FAQ: Counting Techniques Question: Drawing 5 specific cards out of a deck

1. How many different combinations of 5 cards can be drawn from a standard deck?

There are 2,598,960 possible combinations of 5 cards that can be drawn from a standard deck of 52 cards.

2. What is the probability of drawing a specific hand of 5 cards from a deck?

The probability of drawing any specific hand of 5 cards from a deck is 1 in 2,598,960 or approximately 0.0000385%.

3. How does the order of the cards affect the number of possible combinations?

The order of the cards does not affect the number of possible combinations. As long as the same 5 cards are drawn, it is considered one combination.

4. Can a single card be included in multiple combinations of 5 cards?

No, in a combination of 5 cards, each card can only be included once. For example, if a hand contains the Ace of Hearts, it cannot also contain another Ace of Hearts.

5. How does the number of cards in a deck affect the number of possible combinations?

The number of cards in a deck directly affects the number of possible combinations. As the number of cards increases, the number of possible combinations also increases exponentially.

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