Four cards are randomly drawn from a pack of 52 cards. Find

Question

Four cards are randomly drawn from a pack of 52 cards. Find the probability that all four cards are of different denominations?

My solution:

OMEGA = 52 x 51 x 50 x 49
A = 52 x 48 x 44 x 40

P(A) = A/OMEGA = 0.68

Official solution:

OMEGA: (52 4)
A: (4 1) x (13 4)
P (A) = 0.01

I think the official solution is wrong.

Bunuel · Accepted Answer

I think you are right: \frac{4^4*C^4_{13}}{C^4_{52}} would be the probability of getting 4 cards of different denominations out of 52. # of ways of to choose 4 different denominations out of 13 - C^4_{13} ; Multiplying by 4^4 as each card from the above string can be of four suits ( 4*4*4*4=4^4 ); Total # of ways to choose 4 cards out of 52 - C^4_{52} . Or another way: We can choose any card for the first one - \frac{52}{52} ; Next card can be any card but 3 of the denomination we'v already chosen - \frac{48}{51} (if we've picked 10, then there are 3 10-s left and we can choose any but these 3 cards out of 51 cards left); Next card can be any card but 3*2=6 of the denominations we'v already chosen - \frac{44}{50} (if we've picked 10 and King, then there are 3 10-s and 3 Kings left and we can choose any but these 3*2=6 cards out of 50 cards left); Last card can be any card but 3*3=9 of the denominations we'v already chosen - \frac{40}{49} ; P=\frac{52}{52}*\frac{48}{51}*\frac{44}{50}*\frac{40}{49}=\frac{4^4*13*12*11*10}{52*51*50*49} - the same answer as above. Hope it helps. Please post PS questions in the PS subforum: gmat-problem-solving-ps-140/ Please post DS questions in the DS subforum: gmat-data-sufficiency-ds-141/ No posting of PS/DS questions is allowed in the main Math forum.

nonameee · Answer

I sorted it out. My solution is wrong: 52 x 48 x 44 x 40 means that this set will actually contain cards of the same denomination. The official solution is IMO still incorrect since A should be equal to: (4^4) x (13 4).

Your opinion would be greatly appreciated.

nonameee · Answer

Bunuel, thank you for your answer.

I sorted it out. My solution is wrong: 52 x 48 x 44 x 40 means that this set will actually contain cards of the same denomination. T

Wanted just to correct myself. The above argument is wrong since: 52 x 48 x 44 x 40 is a case of variations without repetition. Therefore, the set won't contain cards of the same denomination.

Jacksparrow1906 · Answer

Bunuel, thank you for your answer.

Why wouldn't the probability be 1 * 13/51 * 13/50 * 13/49 ? We know there are 4 suits, and the probability of getting any card in one suit is 13/# of cards in the deck.

Jacksparrow1906 · Answer

In your solution, you found the # of ways to choose 4 different denominations out of 13, I believe that is incorrect. Each suit(denomination) has 13 cards, we're looking for the # of ways to draw 1 card of a different suit each time, up to 4 draws. Correct me if I'm wrong.

Bunuel · Answer

Yes you are wrong.

First of all suit and denomination are not the same: there are four suits (Clubs, Diamonds, Hearts, and Spades), and 13 different denominations (Ace, two through ten, Jack, Queen, and King).

Next, even if you are counting  the probability that the 4 cards you draw are of different suits , 1 * 13/51 * 13/50 * 13/49 won't be the correct answer.

Correct answer would be  P=\frac{(C^1_{13})^4}{C^4_{52}} .

Or another way:  P=4!*\frac{13}{52}*\frac{13}{51}*\frac{13}{50}*\frac{13}{49}  (the same probability as above), we are multiplying by 4! as the favorable scenario of Clubs, Diamonds, Hearts, and Spades (CDHS) can occur in 4! # of ways, which is basically the # of permutaions of 4 letters CDHS.

Hope it helps.

P.S. This question is beyond the scope of GMAT, so I don't recommend to spend too much time on it.

Jacksparrow1906 · Answer

Thanks for the explanation, makes perfect sense now

ssgomz · Answer

confused

why is it multiplied by 4^4 and not 4!? if the first card is taken from 1 suit the second can be from the remaining 3 because 1 suit will have only one card of one denomination....?

Bunuel · Answer

We need 4 cards of different denomination, but these 4 cards don't have to be of the different suits.

Consider this:  C^4_{13}  is the # of ways to choose 4 cards of different denomination. For example one particular case from  C^4_{13}  would be:  Jack - Queen - King - Ace .

Now,  each card  from this case ( Jack - Queen - King - Ace ) can be of 4 suits: Clubs, Diamonds, Hearts, and Spades. So taking into account the suits the case  Jack - Queen - King - Ace  can be formed  4*4*4*4=4^4  # of times. As there are  C^4_{13}  # of cases like  Jack - Queen - King - Ace , so total # of ways to form 4 cards of different denomination is  C^4_{13}*4^4 .

Hope it's clear.

ssgomz · Answer

Thank you for the detailed explanation. It is clear now

narisipalli · Answer

How did  \frac{52}{52}*\frac{48}{51}*\frac{44}{50}*\frac{40}{49}  become equal to  \frac{4^4*13*12*11*10}{52*51*50*49}  here?

Take 4 common from the numerator and the result would be  4*\frac{13*12*11*10}{52*51*50*49} .

Bunuel · Answer

Hi,

\frac{52}{52}*\frac{48}{51}*\frac{44}{50}*\frac{40}{49}=\frac{4*13}{52}*\frac{4*12}{51}*\frac{4*11}{50}*\frac{4*10}{49}=\frac{4^4*13*12*11*10}{52*51*50*49} .

Hope it's clear.

virupaksh2010 · Answer

I think we we should not multiply it with 4 since we are not finding number of ways just probability so order doesnot matter..

13/52 * 13/51*13/49*13/48 should be the answer ...

Please correct me if i am wrong.

Bunuel · Answer

Your approach is not correct:
First of all  Probability \ of \ an \ event =\frac {# \ of \ favorable \ outcomes}{total \ # \ of \ outcomes} , so we do need to find the # of ways and if order matters for # of ways of an event, then order matters for probability too.

Answer for the original question  is  C^4_{52}=\frac{4^4*13*12*11*10}{52*51*50*49}  ( so we are not multiplying 13/52 * 13/51*13/49*13/48 by 4 we are multiplying by 4^4), please see the solution in my first post.

If you are referring to the  different question  we've considered ( the probability that the 4 cards you draw are of different suits ), the answer for it is still not 4*13/52 * 13/51*13/49*13/48 it's  P=\frac{(C^1_{13})^4}{C^4_{52}}=4!*\frac{13}{52}*\frac{13}{51}*\frac{13}{50}*\frac{13}{49} , so again we are not multiplying by 4 we are multiplying by  4!=1*2*3*4=24 , please see the solution for this question in my second post.

For more on this issue please check Probability and Combinatorics chapters of Math Book (link in my signature).

Hope it helps.

GMATBusters · Answer

Hi
My approach to this question is like this...
first card can be any card , so the probability of first card = P(1) =52/52 =1
2nd card should be from suit different from first draw, so no of favourable choice = 52-13 = 39 ; P(2) = 39/51.
3rd card should be from suit different from first & second draw, so no of favourable choice = 52-(2*13) = 26 ; P(3) = 26/50
4th card should be from suit different from 1st, 2nd and 3rd draw, so no of favourable choice = 52-(3*13) = 13 ; P(4) = 13/49.

Required probability = 1*(39/51)*(26/50)*(13/49).

Timebomb · Answer

Hi

I understand the solution , but I was trying to go binomial rather than having a straight 4^4 .

If anyone could help.

Hearts spade club diamond 
 4. 0. 0. 0. ( Only 1 way to get all 4 denominations from hearts)
Therefore coefficient of 4c4 should be 1.

Hence the expansion should be like (1+3)^4
Coefficient of 4C3 should be 3.

Hearts spade club diamond 
 3. 1. 0. 0.
 3. 0. 1. 0.
 3. 0. 0. 1.
Three cases .. tilL now it makes sense as coefficient should be 3.
Now here is the point where I have issues. Thee coefficient should be 9 but I can get only 6 ways where there are two different denominations from heart and 2 different denominations from other 3 suits namely

Hearts spade club diamond 
 2. 2. 0. 0.
 2. 0. 2. 0. 
 2. 0. 0. 2.
 2. 1. 1. 0.
 2. 1. 0. 1.
 2. 0. 0. 1.

What am I missing here?

Posted from my mobile device

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