Drawing ten cards from 52 cards

In summary, the conversation discussed the total number of possibilities for drawing ten cards from a deck of 52 cards. The first case involved drawing 7 clubs, 1 heart, and 2 diamonds, resulting in 1740024 possibilities. The second case involved drawing 8 clubs, 1 heart, and 1 diamond, resulting in 217503 possibilities. However, there was confusion about the answer for the second case, which was stated to be 117117. The conversation also mentioned using factorization to reverse-engineer the answer and mentioned the new snackers and extra-crispy recipe.
  • #1
KFC
488
4
Hi there,
I am thinking a question from a book about drawing ten cards from 52 cards. The first case is drawing ten cards, having 7 cards of a club, 1 card of a heart, 2 cards of a diamond. The order doesn't matter, so the total number of possibilities is

C(13, 7)*C(13, 1)*C(13, 2) = 1740024

In the second case, if I need to draw ten cards, having 8 cards of a club, 1 card of a heart, 1 card of a diamond, so the total number of possibilities is

C(13, 8)*C(13, 1)*C(13, 1) = 217503

but I was told that for the second case, the answer should be 117117. Do you know how they get the number 117117 or in what case will this number come up?
 
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  • #2
Your answer seems fine to me. One way of seeing where the answer may have come
from is by factoring it:

117117=7.9.11.132 , sometimes that helps reverse-engineer the answer.

BTW, I love the new snackers and the extra-crispy recipe. You're still doing it right.
 
Last edited:
  • #3
Bacle said:
Your answer seems fine to me. One way of seeing where the answer may have come
from is by factoring it:

117117=7.9.11.132 , sometimes that helps reverse-engineer the answer.

BTW, I love the new snackers and the extra-crispy recipe. You're still doing it right.

Thank you for your hint. I was told by someone that though C(13, 8)*C(13, 1)*C(13, 1) is a right number, but 117117 may be used by some card simulator. I am thinking the 13*13 is the way to deal one heart and one diamond, but for the first three number 7, 9 and 11, I still confuse how this three numbers associate to deal cards of club.
 

1. How many possible combinations can be drawn from a standard deck of 52 cards?

There are 52!/(10!(52-10)!) = 52!/10!42! = 158,200,242 combinations that can be drawn from a standard deck of 52 cards when drawing ten cards.

2. What is the probability of drawing a specific hand, such as a royal flush, when drawing ten cards from a deck of 52?

The probability of drawing a specific hand, such as a royal flush, when drawing ten cards from a deck of 52 is 4/158,200,242, which is approximately 0.000000025%. This is because there are only 4 possible royal flush combinations out of 158,200,242 total combinations.

3. How does the probability of drawing a specific hand change if some cards have already been drawn?

The probability of drawing a specific hand changes as cards are drawn because the number of possible combinations decreases. For example, if one card has already been drawn, there are only 51 cards left to draw from instead of 52, making the probability slightly lower.

4. What is the expected value of the number of face cards when drawing ten cards from a deck of 52?

The expected value of the number of face cards when drawing ten cards from a deck of 52 is 10*3/13 = 2.31. This means that on average, we can expect to draw approximately 2-3 face cards when drawing ten cards from a deck of 52.

5. How does the number of possible combinations change if the order of the cards matters?

If the order of the cards matters, the number of possible combinations increases significantly. With a standard deck of 52 cards, the number of possible combinations when drawing ten cards is 52!/42! = 52*51*50*49*48*47*46*45*44*43 = 3,628,800, which is over 22 times the number of combinations when the order doesn't matter.

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