If I had to get a total of at least 18 from dice rolls, how do I figure out how many variations of dice rolls?

Question

If I had to get a total of at least 18 from dice rolls, how do I figure out how many variations of dice rolls?

Answer 1

You can figure out a recursive relationship.

Let N(x) be the number of ways that you can roll x.

The number of ways to roll at least 18 means rolling a 17 before and 1-6, a 16 before and 2-6, a 15 before and 3-6, a 14 before and 4-6, a 13 before and 5-6, and a 12 before and 6 equals

6N(17)+5N(16)+4N(15)+3N(14)+2N(13)+1N(12)

To determine N(x), for x >= 7, you can roll 1-6 and x - (1-6) previously

Thus, N(x) = N(x-1)+N(x-2)+N(x-3)+N(x-4)+N(x-5)+N(x-6) x >= 7

For 1 <= x <= 6, to make the notation easier, let N(0)=N(-1)=N(-2)=N(-3)+N(-4) = 0

Then, N(x) = 1 + N(x-1)+N(x-2)+N(x-3)+N(x-4)+N(x-5) 1 <= x<= 6

Thus, the values of x and N(x) for 1 <= x <= 7 are, writing as (x, N(x)),

(1,1); (2,2); (3,4); (4,8); (5,16); (6,32); (7,63); (8,125); (9,248); (10,492); (11,976); (12,1936); (13,3840); (14,7617); (15,15109); (16,29970); (17,59448)(1,1); (2,2); (3,4); (4,8); (5,16); (6,32); (7,63); (8,125); (9,248); (10,492); (11,976); (12,1936); (13,3840); (14,7617); (15,15109); (16,29970); (17,59448)

and the number of ways to get at least 18 = 6N(17)+5N(16)+4N(15)+3N(14)+2N(13)+1N(12) =

599441