The Probability of High Dice

So my bother wanted to know. What is the probability of getting a particular value on a number of different dice. With one catch. He wanted to know how many dice would have to be thrown to be almost positive to get the highest face value to be a 6, or no bigger than a 5, or a 4 etc. In essence when r dice are thrown, what is the probability of getting no number greater than k. 

As he found, this is fairly easy to do by hand for small numbers of dice, about 3 or less. However after three it's almost impossible, without destroying a small forest, to list all possibilities of combinations for say 4, 5 or more dice. The catch is that at least one of the dice has to attain the desired value and the rest can be equal or smaller than that. Initially I didn't realize this contraint and simply added probabilities of every possible combination of dice including triples of that value. Although we did come up with a stack of other interesting setups, and a number of cool points, we finally got the following:

Once you had the setup right it isn't to hard. Essetially all possible combinations of the dice, minus the combinations that don't include k, the highest number. The interesting point though, is that although you are more likely to get a 6, good for initiative addition or something in the game, the probability of getting the smaller numbers drops of dramatically as the number of dice increases. producing a rather cool graph. Which for the math nerds contains a saddle point if you consider the numbers on the real line (i.e. less than 1 on a die)

From left to right we have r, the number of dice, increasing from 1 to 20; From front to back is k, the maximum value on the dice, increasing from 1 to 6. The upshot, and possible cool rule to add is what we dubbed the Clouseau multiplier. 

A advanced experienced character with 10 dice to roll has a whopping 83% chance of rolling a 6 and a 13% chance of a 5, so we can be assured that the character will score a 5 or a 6. Unlike a lowley newbie with only a 18% of getting a 6, or any number for that matter. However, there is almost 0 possibility of getting all ones if you roll 10 dice. So if a player does succeed in getting 1 as their highest value make it a critical hit. "What does "self destruct" mean?" 

And for those that are awed by those rare occasions. Lets say your character is super powerful and has managed to get to level 42 where you can roll a total of 21 six sided dice. The probability that you get at least one 6 is only.. wait for it... 98%. It's high, but that means that once every 50 games you won't get a single 6. Which is probably every day!