Applying Pascal's Triangle to 16-spot tickets

Apr 29, 2008 7:00 PM

Keno Lil | This week weíll apply the principles of Pascalís Triangle to tickets with three different sized groups on them.

For an example, letís take the 16-spot ticket with two groups of 3, three groups of 2, and four kings. If you remember from last week, we use the second row of the triangle to break out the 3-spots: A 1-way 6, a 2-way 3, and a 1-way "nothing."

We use the third row (because there are three groups of two) to break out the deuce ways: A 1-way 6, a 3-way 4, a 3-way 2, and a 1-way "nothing."

For the same reason, we use the fourth row to break out the 1-spot ways: A 1-way 4, a 4-way 3, a 6-way 2, a 4-way 1-spot, and a 1-way "nothing."

In our keno shorthand we write these breakouts thusly:

1/6
2/3
1/0
1/6
3/4
3/2
1/0
1/4
4/3
6/2
4/1
1/0

 Proceeding as we did last week, we take the first two breakouts and arrange them on a grid, multiplying the numerators and adding the denominators:

1/6
2/3
1/0
 
1/6 
1/12
2/9
1/6
3/4
3/10
6/7
3/4
3/2 
3/8
6/5
3/2
1/0 
1/6
2/3
1/0

The next step is to construct a second grid, using the results of the first calculation for the top row, and the 1-spot breakout for the vertical column:

The 1-spot Breakout
1/12
3/10
2/9
3/8
6/7
2/6
6/5
3/4
2/3
3/2
1/0 
1/4 
1/16
3/14
2/13
3/12
6/11
2/10
6/9
3/8
2/7
3/6
1/4
4/3 
4/15
12/13
8/12
12/11
24/10
8/9
24/8
12/7
8/6
12/5
4/3
6/2
6/14
18/12
12/11
18/10
36/9
12/8
36/7
18/6
12/5
18/4
6/2
4-1
4/13
12/11
8/10
12/9
24/8
8/7
24/6
12/5
8/4
12/3
4/1
1/0
1/12
3/10
2/9
3/8
6/7
2/6
6/5
3/4
2/3
3/2
1/0

Including the 16-spot, but ignoring the now meaningless 1/0, we find the expected 511 total ways, once they are summed up. Easy, once you know the technique!

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 [email protected]. Well, thatís it for now. Good luck! Iíll see you in line!