Posted by on 11/25/2010
Well it started when the lotto jackpot rises to as much as 500M pesos on 6/55. We had an argument at home as to how many is the possible combination in the said game. Being a math addict, I calculated it and it was about 20B in my computation meaning that you would need a least 400B to win (since it is 20Php for every combination). But then they we were telling me I was wrong because a personnel from the PCSO said on the TV that it was just around a million. So I ask myself, how does it happen?

Fortunately, we (actually 'they') had an argument too in the office on the similar issue, and being engineers, we do seek for an answer in a logical manner. To resolve this let's first check the description of combination and permutation:

- If the order doesn't matter, it is a Combination.
- If the order does matter it is a Permutation.
Using that description, we might conclude that lotto uses Combination specifically a Combination with no-repetition .

The formula I used to 20B is something like this:

P(n,r) = n! / (n - r)!

where:
n = number of choices
r = number of positions to fill in

The equation is true only that this works only for Permutation. Using this you will get 20,872,566,000 for the number of permutations in 55 numbers to form a set of 6 if order does matter.

But the Lotto uses Combination...

To solve the combination you would need to divide it by the reciprocal of the factorial of the number of positions making the equation as:

C(n,r) = n! / (r!(n - r)!)

where:
n = number of choices
r = number of positions to fill in

This is the correct way to solve the outcome for the number of combinations. Using this the actual combination would be 28,989,675 only. Meaning you need to have at least Php579,793,500 to win.

The question would be, why isn't there a winner yet? Well basically, this is still risky even if you have the money of Php579,793,500, the probabability that you may not be the sole winner is high thinking that there are about a million of Filipinos who are also betting on the game. :D