Bill has a small deck of 12 playing cards made up of only 2 suits of 6

Bunuel, very nice explanation. I believe, the second method can be derived logically and the first one with some knowledge of equations in permutations and combination. Thanks once again.

In method 1, the order of picking matters
In method 2, when you use the C(n,r) formulae, the order of picking does not matter
As a result method 2 will always yield a lower number of ways than method 1
Eg. In method 1, you will count the set Suit1No1, Suit1No2,Suit2No3,Suit2No4, 4! times, one for each ordering between the 4 cards that exists, whereas in method 2, you will count it only once.

It is essentially the difference between P(n,r) and C(n,r)

This subtlety is present in a lot of probability questions and must be clarified in the Q. The above question is in MGMAT (I got it on my CAT), but it fails to tell you if the 4 cards that are picked are picked simultaneously (i.e. order does not matter) or one by one. In my opinion such questions are ambiguous and the answer is dependent on the assumption you make. Unfortunately there are more Qs like this one on the MGMAT CATs, just something to be weary of.

Whether you pick the four cards simultaneously, or pick them one at a time (without replacement) doesn't actually matter if you are finding a probability; the two situations are mathematically identical. You can see this intuitively by thinking of taking hold of four cards in the deck first. If you take them all out at the same time, or if there is a nanosecond between your removing each, why would the probability that you get a pair be affected? It won't be, so the 'ambiguity' you suggest is present in such questions is no ambiguity at all.

You can see that either perspective will give you the same answer, though it's easier to illustrate with a simpler example. Say you have 3 red marbles and 4 blue marbles in a bag, and you pick two (either simultaneously, or without replacement - it's the same thing), and you want to find the probability of picking two red marbles. If we look at the problem as though we are picking marbles one at a time, we have 3*2 ways of picking two reds, and 7*6 ways of picking two marbles, so the probability would be 3*2/7*6 = 1/7. If we look at the problem as though we're picking two marbles simultaneously, we have 3C2 ways of picking two red marbles and 7C2 ways of picking two marbles, so the probability would be 3C2/7C2 = 1/7. So when you stick your two hands in the bag and grab two marbles, it doesn't matter if you lift your two hands out at the same time, or take them out one at a time; the probability is the same.

Note though that you need to be consistent in the calculation - if you assume order matters when you calculate the numerator, you must also assume order matters when you calculate the denominator.

You are right, what I should have said is that the ambiguity occurs if you are solving the combinatorial problem, where you are counting outcomes.
In calculating probability it is irrelevant as the ordering terms will cancel themselves out.

In the question mentioned, the problem is if you calculate the number of relevant ways using one assumption and the total number of ways using another assumption

There is 2 major problems with this :

(a) By choosing 2 cards from one suit and 2 from the other, you are forcing that the choice has exactly 2 cards from each suit, which is not true

(b) Even if you had to calculate no of ways to pick 4 cards such that no pairs and 2 cards from each suite, this is wrong, because it needs to be multiplied by 2 [2 ways to choose which is the "first suite"]

This calculation is not really too lengthy. Although this is a difficult question, it is fair game for the GMAT at higher score levels.

Required probability = 1 -prob(no pair of cards have the same value)
= 1 - [(12x10x8x6/12x11x10x9)]
= 1 - 48/99
= 17/33

Option (C)

Bill has a small what?

sorry for that joke ))) Actually, this question was discussed trillions times. that is why I cant do anything, but smile )

https://gmatclub.com/forum/bill-has-a-small-deck-of-12-playing-cards-made-up-of-only-2-suits-of-6-cards-each-96078.html

P.S. my advise- use google search and ur life will be brighter

actually i don't care what has Bill and what size it is - but it took me more than two min to find it out
compared to trillion +-1 doesnt make any difference )
but ,anyway, thank you for an advice - a quite good one!

The way that is given here I understand.
pairs-twins-and-couples-103472.html#p805725

But I am trying to do in another way , wonder what I am doing wrong.
at least one pair means either one pair or two pairs
so lets suppose we have A1 A2 A3 A4 A5 A6 AND B1 B2 B3 B4 B5 B6

I understand the 1 - p( opposite event )

but just for understanding sake , if I tried the direct way , how could it be done.

P( One pair) + p( two pairs )


one pair

6C1*5C2(4!/2!)= 720

6C1 = ways to select the one card from 6 which will form the pair
5C2 = select two different cards from the 5 remaining to form the singles * This I think this is not correct , what would be correct *
4!/2! = permutations of 4 letters where two are identical


for two pairs

6C2 * 4!/2!2! = 90

6C2 = ways to select the two cards which will form the pair .
4!/2!2! = ways to arrange the 4 letters where 2 are same kind and another 2 are same kind .

hence total ways for at least one pair -> 6C1*5C2(4!/2!) + 6C2 * 4!/2!2! = 810 ( what is wrong here)
however this will not the required answer, can anybody please correct my logic and show me how to approach this direct way ?
total ways 4 cards can be selected 12C4 = 495 hence I am getting a probability more than one which is not possible

Would highly appreciate any help.

One pair - should be 6C1*5C2*2*2 = 6*10*4 = 240. Choose one pair out of 6, then two single cards from two different pairs. You have two suits, so for the same number two possibilities.
Two pairs - just 6C2=15.
Total - 255
Probability 255/495 = 17/33.

10C2 also includes remaining pairs of cards with the same number, so choosing 2 out of 10 does not guarantee two non-identical numbers.
Another problem is that 495 represents the number of choices for 4 cards, regardless to the order in which they were drawn. Then you should consider the other choices accordingly, and disregard the order in which they were chosen.

Just edited my original post
Meant to take 5C2 not 10C2,

so my doubt

one pair

6C1*5C2(4!/2!)= 720

6C1 = ways to select the one card from 6 which will form the pair
5C2 = select two different cards from the 5 remaining to form the singles * This I think this is not correct , what would be correct *
4!/2! = permutations of 4 letters where two are identical


for two pairs

6C2 * 4!/2!2! = 90

6C2 = ways to select the two cards which will form the pair .
4!/2!2! = ways to arrange the 4 letters where 2 are same kind and another 2 are same kind .

Responding to a pm:

There is a difference between this question and the other one you mentioned.

In that question, you were using one digit twice which made them identical. Here the two cards that form the pair are not identical. They are of different suits. Also, here, you don't need to arrange them. You can assume to just take a selection while calculating the cases in the numerator and the denominator. The probability will not get affected. In the other question, you needed to find the number of arrangements to make the passwords/numbers and hence you needed to arrange the digits.


Hence, in this step, 6C1*5C2(4!/2!), it should be 6C1*5C2*2*2 instead.

6C1 = ways to select the one card from 6 which will form the pair
5C2 = select two different values from the 5 remaining to form the singles *This is correct *
2*2 = For each of the two values, you can select a card in 2 ways (since you have 2 suits)

Similarly, 6C2 * 4!/2!2! should be 6C2 only to select 2 pairs out of 6.

Probability = 255/495 = 17/33


Note: You can arrange the cards too and will still get the same probability. Just ensure that you arrange in numerator as well as denominator.

Only one pair = 6C1*5C2*2*2 * 4! (you multiply by 4! because all the cards are distinct)
Both pairs = 6C2 * 4! (you multiply by 4! because all the cards are distinct)

Select 4 cards out of 12 = 12C4 * 4! (you multiply by 4! because all the cards are distinct)

Probability = 255*4!/495*4! = 17/33

one pair
6C1*5C2(4!/2!)= 720

NO
Choose one pair out of 6 - 6C1 - and you don't care for the order in which you choose the two cards
Choose two pairs out of the remaining 5 pairs - 5C2 - and then, from each pair, you have 2 possibilities to choose one of them, therefore 5C2*2*2
Here you stop! Don't care about any order. Period.

for two pairs
6C2 * 4!/2!2! = 90

NO
You choose two pairs - 6C2 - and you stop here. From each pair you take both cards, and don't care about any order.

Responding to a pm:

How do we obtain \(6C4*2^4\)?

Think of what you have:
6 cards numbered 1 to 6 of 2 different suits. Say you have 1 to 6 of clubs and 1 to 6 of diamonds. A total of 12 cards.

You want to select 4 cards such that there is no pair i.e. no two cards have the same number. This means all 4 cards will have different numbers, say, you get a 1, 2, 4 and 6. The 1 could be of clubs or diamonds. The 2 could be of clubs and diamonds and so on for all 4 cards.

This means you must select 4 numbers out of the 6 numbers in 6C4 ways.
Then for each number, you must select a suit out of the given two suits.
That is how you get 6C4 * 2*2*2*2

This is the total number of ways in which you will have no pair i.e. two cards of same number.

Hi Bunnel

Can you please correct me where am I going wrong in the below approach

1-P(no pair)
P(no pair) = (10x8x6)/12C4

Assuming that the first card can be any among them, let it be 1,2,3,4,5 or 6. Its a fixed value. So the number of ways of selecting it is one. After that the number of ways of selecting the second card is 10 and so on.

Please correct me where i am going wrong.

Thanks!!

You are mixing concepts. You have 12 distinct cards. There are two different methods of solving this:

Method 1: The "number of ways" of selecting the first card is 12, not 1. The number of ways of selecting the second card is 10. The number of ways of selecting the third card is 8 and the number of ways of selecting the fourth card is 6.
Now you have the 4 cards such that there are no pairs.
What are the total number of ways of selecting 4 cards? They are 12*11*10*9
Required Probability = 1 - (12*10*8*6)/(12*11*10*9) = 17/33

Note that you have arranged cards in first, second, third and fourth places. It doesn't matter as long as you arrange them in the denominator as well which you did by using 12*11*10*9 and not 12C4.

Method 2: 1 is the probability of selecting the first card such that there are no pairs. It can be any card so probability of picking it is 1. In this case the second card probability is 10/11 and so on...
Required Probability = 1 - 1*(10/11)*(8/10)*(6/9) = 17/33

Hi Guys ,can someone please tell me why this method would be incorrect?
To select 4 cards from 2 different suits with each having different values ,I could go the following ways
a)Select 2 each from Suit A and Suit B -> 6c2*4c2(To get rest 2 cards from the 4 different values of the other suit)
b)3 from A and 1 from B-> 6c3*3c1
c) 1 from A and 3 from B->6c1*5c3
d)All four from A -> 6c4
e)All four from B->6c4
when I add them all up I get the correct answer but I'm not sure if this is the correct approach..Please guide me!! Thanks