What are the Odds? Part 1

First, there's a $\dfrac{20}{52}$ chance of drawing any of the cards necessary for any Royal Flush with the first card. Thereafter, the chances of drawing the rest of what's needed to complete the Royal Flush is $\dfrac {4 \cdot 3 \cdot 2 \cdot 1} {51 \cdot 50 \cdot 49 \cdot 48}$ . Mutliply them together to get $\dfrac{1}{649740}$

You first need to find how many combinations of 5 cards are possible. To find that, multiply 52 (the number of possibilities for the first card) by 51 (the number of possibilities for the second card) by 50 by 49 by 48. Remember, the order of the cards doesn't matter, so you then have to divide by 5!, which is 120. You end up with 2,598,960, the number of possible combinations for 5 cards of a 52 card deck. Since there are 4 ways you can get a royal flush (clubs, diamonds, hearts, and spades), the odds are 4 in 2,598,960, which simplify to 1 in 649,740.

Theres only 4 possible combinations out of the total of ${52 \choose 5}$ combinations. Hence the odds are 1 in $\frac {2598960}{4} = 649740$ .

So the next time you get a royal flush, consider yourself pretty lucky.