A few hundred years ago a mathematician named Blaise Pascal lived in France. Because he had an interest in probability and gambling, many of his discoveries have pertinence to us today. Among his discoveries is Pascal’s Triangle.
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:
You can continue this process indefinitely, but for our purposes ten lines is enough. What does this have to do with keno? Well, Pascal’s Triangle illustrates quite a few mathematical and statistical relationships. Among these are the fact that each row of the triangle displays the coefficients of the binomial expansion, (a+b) to the nth power, where n is the nth row of the triangle. Our interest lies in the fact that each row of the triangle tells us how many ways there are on any keno ticket that has only one size group on it.
For instance, suppose that we have a king ticket, a ticket comprised of only groups of one (kings.) If we have eight kings (eight groups of one) we will have a way ticket with a one way eight, eight way seven, 28 way six, 56 way five, 70 way four, 56 way three, 28 way 2, and an 8 way one spot. By looking at the eighth row of the triangle, you can read these numbers directly across the triangle. This will also work with a deuce way ticket (a ticket comprised of two spots only), or any other keno ticket that has only one size group on it. As another example, consider the ticket comprised of five groups of four. If we look at the fifth row of the triangle, we can see that there will be one twenty spot ticket, five 16 spots, 10 12 spots, 10 eight spots and 5 four spots.
Pascal’s Triangle has another connection to keno. If you sum each row, the sum of the numbers in each row is equal to two to the nth power, where n is the row number. For instance the sum of the fourth row is 16, which is two to the 4th power. If you subtract one from this number, you will have the total number of ways on a keno ticket with four groups. Thus the formula for the total number of ways on a keno ticket is (two raised to the nth power) minus one, where n is the number of groups on the ticket. For instance, if we sum the numbers in the 8th row of the triangle, we’ll have 256, and we know that on a keno ticket with eight kings, there will be a total of 256-1= 255 total ways on the ticket.
Well, that’s it for now. Good luck! I’ll see you in line!