$52$ ताशों की एक गड्डी से $4$ पत्तों को चुनने के तरीकों की संख्या क्या है ? इन तरीकों में से कितनों में से कितनों में

चार पत्ते एक ही प्रकार $(suit)$ के हैं ?

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 four suits: diamond, club, spade, heart and there are $13$ cards of each suit. Therefore, there are $^{13} C _{4}$ ways of choosing $4$ diamonds. Similarly, there are $^{13} C _{4}$ ways of choosing $4$ clubs, $^{13} C _{4}$ ways of choosing $4$ spades and $^{13} C _{4}$ ways of choosing $4$ hearts. Therefore

The required number of ways $=\,^{13} C _{4}+^{13} C _{4}+^{13} C _{4}+^{13} C _{4}$

$=4 \times \frac{13 !}{4 ! 9 !}=2860$