What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these
two are red cards and two are black cards,
What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these
two are red cards and two are black cards,
There will be as many ways of choosing $4$ cards from $52$ cards as there are combinations of $52$ different things, taken $4$ at a time. Therefore
The required number of ways $=\,^{52} C _{4}=\frac{52 !}{4 ! 48 !}=\frac{49 \times 50 \times 51 \times 52}{2 \times 3 \times 4}$
$=270725$
There are $26$ red cards and $26$ black cards. Therefore, the required number of ways $=^{26} C _{2} \times^{26} C _{2}$
$=\left(\frac{26 !}{2 ! 24 !}\right)^{2}=(325)^{2}=105625$
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- [IIT 2017]