Five cards are selected randomly from a standard deck of $52$ cards. Find the probability that all of them are of different rank.
My try:
The first card can be picked in $52$ ways. Now ignore the other three cards of this rank. Now pick second card which can be done in $48$ ways. Third in $44$ ways. Fourth in $40$ ways and fifth in $36$ ways. Since the order of selection is not important, the total ways is: $$\frac{52 \times 48 \times 44 \times 40 \times 36}{5 !}$$
The required Probability is: $$p=\frac{\left(\frac{52 \times 48 \times 44 \times 40 \times 36}{5 !}\right)}{\binom{52}{5}}=\frac{2112}{4165}$$
Is this correct way?
Indeed, that is correct.
The probability for obtaining five from thirteen ranks with each of one from four suits, when selecting five from fifty two cards is:$$\dfrac{\dbinom{13}5\dbinom{4}1^5}{\dbinom{52}{5}}=\dfrac{(13\cdot 4)(12\cdot 4)(11\cdot 4)(10\cdot 4)(9\cdot 4)}{52\cdot51\cdot50\cdot 49\cdot 48}$$