# Let me (help) count the ways

Dec 18, 2006 6:16 AM

A frequent question is, "How many ways are on this ticket?"

There is an easy formula to determine the total number of ways on a ticket, provided that the ticket has 15 numbers or less. The formula is based entirely on the number of groups, not the number of spots or the size of the groups.

This series of numbers, which is constructed by starting with a 1, doubling and adding 1 at each step, is the series which gives you the total number of ways on a keno ticket.

Here it is: 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, ... etc.

Take this ticket for example: 12 numbers, grouped 3-3-3-1-1-1. You can see by inspection that the ticket has a total of six groups. By counting up the above series 6 times, 1, 3, 7, 15, 31, and 63, we know that the ticket has a total of 63 ways on it. Let’s check! Working it out on paper, we can find a 1-way-12, 3-way-11, 3-way-10, 4-way-9, 9-way-8, 9-way-7, 6-way-6, 9-way-5, 9-way-4, 4-way-3, 3-way-2, and 3-way-1 for a total of 63 ways!

Or take this one for example: 9 numbers, grouped 2-2-2-1-1*1. Counting up the series six times again (because this ticket also has six groups) gives us the answer of 63 total ways. Working this one on paper shows a 1-way-9, 3-way-8, 6-way-7, 10-way-6, 12-way-5, 12-way-4, 10-way-3, 6-way-2, and 3-way-1, for a total of 63 ways.

Try this one: 8 numbers, grouped 2-2-1-1-1-1. Since there are six groups, the above series tells us that there will be a total of 63 ways! Indeed, checking the breakdown of the ticket provides us with a 1-way-8, 4-way-7, 8-way-6, 12-way-5, 14-way-4, 12-way-3, 8-way-2, and 4-way-1 for a total of 63 ways.

It is true, that ANY ticket with 6 groups on it will have a total of 63 ways! (As long as there are fifteen numbers or less.) The limitation to fifteen numbers is required because in most keno games, fifteen numbers are the most you can play on any one way. If you could play 16, 17, 18 spot tickets and up, the limitation would not apply.

Try this one: 10-spots, grouped 1-1-1-1-1-1-1-1-1-1. As you can see, there are ten groups on the ticket, so by counting up the series 10 times, we can see that there must be 1023 ways on the ticket total! Indeed, there is a 1-way-10, 10-way-9, 45-way-8, 120-way-7, 210-way-6, 252-way-5, 210-way-4, 120-way-3, 45-way-2, and a 10-way-1, for a total of 1023 ways!

Go ahead and play around a little with this, and you’ll see that it works quite nicely!

For those of you who remember your high school algebra, the mathematical formula for this series is 2n-1. (Two raised to the nth power, then subtract one.- N represents the number of groups on the ticket.