Well for six cards you start by finding the number of six card combinations in a deck of 52

that is denoted as 52C6 (here is a website that will do the calculations for you http://www.wcrl.ars.usda.gov/cec/java/comb.htm)

so we see there are 20,358,520 six card combinations in a deck of 52.

Next we have to figure out how many of those six card combinations contain a royal flush. We know that there are 4 royal flushes containing 5 cards, each roayl flush could then be combined with any one of the 47 remaining cards in the deck so that gives us 4 X 47 ways to make a royal flush = 188 so the odds are 188 in 20,358,520 which can be expressed as 1 in 108,290.

If you want to calculate for 7 cards you do basically the same thing there are 133,784,560 combinations of 7 cards out of a deck of 52. For each Royal flush will be 2 additional cards. so out of the 47 remaining cards there are 1081 combinations of 2 cards that can be made. Each of the 4 royals can combine with each of the 1081 combinations so there are 4,324 7card combinations containing a royal Flush. 4,324 in 133,784,560 is 1 in 30,940

Now when you get to a draw game it is far more difficult, because obviously it is dependent on what cards the player draws. Once you know what the first cards are you can calculate the chance of drawing to a royal for each combination that a player could hold.