Hey guys little bit stuck on this probability question any help would be great.
"A regular deck of playing cards contains 52 cards, you are randomly dealt 7 from the the 52. What is the probability of you receiving two pairs of face cards of different ranks (e.g. $J-J, \ K-K$) and any three cards whose ranks are different from those of both pairs?"
So far I have worked out that the probability of receiving one pair is ${_3}C_1\times {_4}C_2$ and I think the probability of the 2nd pair will be ${_2}C_1\times {_4}C_2$. I'm a little confused on the 2nd part tho, my guess is that it will be something like ${_{44}}C_3$. I know this will all be over ${_{52}}C_7$.
So my worked out answer is $\Large{\frac{{_3}C_1\times{_4}C_2\times{_2}C_1\times{_4}C_2\times{_{44}}C_3 }{ {_{52}}C_7}}$. If this is slightly incorrect any hints in the right direction would be great.
Two answers
Firstly, you have to take note that order does not matter for choosing the ranks of the pairs.
Secondly, normally, such questions imply exactly two pairs. but the way it is worded, it could mean at least two pairs, which becomes more complex. I'll solve for both.
Exactly two pairs:
After you have chosen the ranks for the pairs, 11 ranks remain for choosing "singles"
$\dfrac{{3\choose 2}{4\choose2}^2{11\choose 3}{4\choose 1}^3}{52\choose 7}$
At least 2 pairs:
To the numerator for exactly two pairs, add the part for exactly 3 pairs :
${3\choose 3}{4\choose 2}^3{10\choose 1}{4\choose1}$