Question

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these, four cards are of the same suit?

• A. $270725,2860$
• B. $270720,2865$
• C. $270724,2869$
• D. None of these

Answer 1

Answer: (A)

$270725,2860$

Answer 2

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 $=\, ^{52}C_{4}$ $= \frac{52!}{4! \,48!} = 270725$ There are four suits: diamond, club, spade, heart and there are $13$ cards of each suit. $\therefore$ The required number of ways $=\, ^{13}C_{4} +\,^{13}C_{4} +\, ^{13}C_{4} +\,^{13}C_{4}$ $= 4\times\frac{13!}{4! \,9!}$ $= 2860$