Keno Lil | Pascal’s triangle may be constructed by taking a piece of paper and writing down the two numbers 1 and 1 side by side. This will be the first line of the triangle:
Subsequent lines are added by starting each line with a 1, and then by adding together the vertices of the triangle diagonally above, and ending each line with a 1 as illustrated here.
We’ve discussed before the application of Pascal’s Triangle to elementary keno calculations, those tickets that have all the same size groups on them. For instance, given a way ticket with 10-spots on it, grouped in five groups of two, we use the fifth row of the triangle, and reading across, we get a 1-way-10, 5-way-8, 10-way-6, 10-way-4, 5-way-2, and a 1-way "nothing." Now the "nothing" means nothing in itself, but it will serve as a place holder in further calculations.
What if the ticket is a mixed group ticket? For instance, suppose it has ten numbers, grouped 3-3-2-2? For this we need to do a two step calculation. First, using the second line for the two groups of three, we find that we have a 1-way-6, a 2-way-3, and a 1-way "nothing." Putting this in a fractional notation, we write:
1/6 2/3 1/0
Of course these are not true fractions in the mathematical sense, just a keno shorthand that we use. We read 2/3 as "2-way-3." Doing the same with the two groups of two, we get:
1/4 2/2 1/0
We next create a grid, with the 3-spot ways across the top, and the 2-spot ways in the leftmost column. We can then calculate the ways on the ticket by multiplying the numerator and adding the denominator to fill the grid, like this:
1/6 2/3 1/0 1/4
1/10 2/7 1/4 2/2
2/8 4/5 2/2 1/0
1/6 2/3 1/0
You will now find all 15-ways on the ticket displayed inside the grid lines, plus the "place holder" (1/0) which we can ignore. Next week we’ll cover tickets with three different sized groups using this method.
If you have a keno question that you would like answered, please write to me care of this paper, or contact me on the web via email at firstname.lastname@example.org. Well, that’s it for now. Good luck! I’ll see you in line!