Question

What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Answer 1

$\displaystyle |\Omega|=\binom{52}{5}=\dfrac{52!}{5!47!}=\dfrac{48\cdot49\cdot50\cdot51\cdot52}{2\cdot3\cdot4\cdot5}=2598960\\ |A|=\binom{13}{5}\cdot4=\dfrac{13!}{5!8!}\cdot4=\dfrac{9\cdot10\cdot11\cdot12\cdot13}{2\cdot3\cdot4\cdot5}\cdot4=5148\\\\ P(A)=\dfrac{5148}{2598960}=\dfrac{33}{16660}\approx0.2\%$