# Rolling dice statistics, probability of rolling a straight

How do I correctly set up an equation to solve for the probability of rolling a straight of X amount. First with 5, then 10 then 15 dice and so on. See image.

So for example: (6 sided die) I would roll 10 dice all at once, order does not matter. What is the probability I would get at least one straight of 3. Straight of 3 meaning a set of one of these numbers 123 or 234 or 345 or 456.

Edit (rephrasing example question):
What is the probability of rolling 1,2,3 or 2,3,4 or 3,4,5 or 4,5,6 out of 10 dice?

I am making a statistics chart that I can pull data from when making cards for a board game. If this can be done in excel that would be helpful too. I am guessing this is a one and done scenario, so I am not too focus on retaining the knowledge.
• The bounty is low!

• What do you believe a fair amount for this would be?

• Your example is not clear

• Well, it would take at least 30 minutes to write a good solution. Offer whatever you think is a fair bounty for a skilled professional to work for half hour on your problem.

• So you roll n die at the same time and you want to compute the probability of say rolling at least one 1, one 2, and one 3 so you have a 123, and you don't care what happens for the remaining (n-3) die?