A probability problem by Trung Le

To draw a red flush we must draw either: (i) any $5$ cards from the suit of hearts, or (ii) any $5$ cards from the suit of diamonds. As each suit is composed of $13$ cards, we have $\dbinom{13}{5}$ combinations of cards from each of options (i) and (ii) that will yield a red flush.

As there are $\dbinom{52}{5}$ possible 5-card draws without restrictions, the desired probability is

$\dfrac{2*\dbinom{13}{5}}{\dbinom{52}{5}} = \dfrac{\dfrac{2*13!}{8!*5!}}{\dfrac{52!}{47!*5!}} = \dfrac{2*13*12*11*10*9}{52*51*50*49*48} =$

$\dfrac{26}{52}*\dfrac{12}{48}*\dfrac{10}{50}*\dfrac{9}{51}*\dfrac{11}{49} = \dfrac{3*11}{2*4*5*17*49} = \boxed{\dfrac{33}{33320}}.$