A Comprehensive Guide to Calculating Lottery Combinations

A Comprehensive Guide to Calculating Lottery Combinations

In this article we will demystify the tricky topic of lottery combinations. We’ve split the article in two, starting with some straightforward answers to questions around lottery combinations, before we dive into a more in-depth look. If you don’t like maths – skip the last 2 sections!

Contents:
What Is A Lottery Combination
Why Are Lottery Combinations Important
Can You Buy Every Possible Lottery Combination?
Why Do People Care About Lottery Combinations?
Are Some Lottery Combinations More Likely To Win Than Others?
How To Calculate Lottery Combinations

What is a lottery combination?

At its simplest, a lottery combination is just the combination of numbers that you pick.

So, if you’re playing the Health Lottery then you choose 5 numbers between 1 and 50 – this is your combination. Read our guide to find out more about how to play the Health Lottery.

Because the order you choose the numbers doesn’t matter, both 1,5,17,34,48 and 5,17,1,48,34 count as the same combination.

Why are lottery combinations important?

At this point you might wonder what all the fuss is about. After all, every number is as likely as every other number to appear in the lottery draw – so what’s so important about the combination that you choose?

Well, the first thing that’s important is that the number of different combinations available affects the odds of winning the lottery. Put simply – the more possible combinations there are, the less likely you are to have chosen the winning one with any single lottery ticket.

Can you buy every possible lottery combination?

Technically, yes – but it’s a really bad idea...

As we’ve already seen, there are a lot of combinations. So, the logistics alone of buying every single one would make it borderline impossible - that’s a lot of runs to your local lottery ticket retailer!

Then, on top of that, you’d actually spend more on the tickets than you could ever hope to win back, which isn’t particularly responsible either. That’s before you consider the fact that someone else could also have a winning ticket, so you’d probably end up splitting the jackpot anyway…  So, you’re better off just buying a ticket and hoping that luck is on your side!

Why do people care about lottery combinations?

Where lottery combinations become more of a hot topic is the idea that choosing certain types of combination increase your chances of winning. Everybody has an opinion on this one. To get to the bottom of it, we’ll have to take a look at the numbers and work out how to calculate lottery combinations…

Are some lottery combinations more likely to win than others?

Short answer: No. Any set of numbers chosen in a lottery is as likely to win as any other set, because all numbers have the same probability of being drawn.

So why do we care? Well, even though any given combination is as likely as any other to appear, lots of players still like to be strategic about the combinations they go for and there’s good reason for this: certain types of combinations have better odds than others:

Combinations of Even and Odd numbers.

Choosing combinations with a balance of both odd and even numbers is a common strategy. This is because the winning numbers in a lottery are more likely to have a balance of even and odd numbers than to have only odd or only even numbers.  

To see why this is, let’s use the Health Lottery as an example:

You choose 5 numbers, so that means you can have the following types of combinations:

5 even 0 odd 2 even 3 odd
4 even 1 odd 1 even 4 odd
3 even 2 odd 0 even 5 odd

 

There are 25 odd numbers and 25 even numbers to choose from, as you select numbers from 1-50. This means that there are more combinations of the form 3-even 2-odd (690,000 in total) than there are 5-even 0 odd (53,130 in total). So, it is more likely that the winning combination of the lottery is of the form 3-even 2-odd than it is of the form 5-even 0-odd.  

Combinations of High and Low numbers.

Another common combination strategy is to choose a balance of high and low numbers. This is because, like we saw in the previous example, it is more likely that the winning numbers in the lottery will have a spread of numbers, rather than being all low or all high numbers.

To see why this is, let’s use the Health Lottery as an example (the maths is exactly the same as the even/odd case we looked at before):

You choose 5 numbers, so that means you can have the following types of combinations:

5 low 0 high 2 low 3 high
4 low 1 high 1 low 4 high
3 low 2 high 0 low 5 high

 

There are 25 low numbers (1-25) and 25 high numbers (26-50) to choose from. This means that there are more combinations of the form 3-low 2-high (690,000 in total) than there are 5-low 0-high (53130 in total). So, it is more likely that the winning combination of the lottery is of the form 3-low 2-high than it is of the form 5-low 0-high. 

How To Calculate Lottery Combinations

To calculate lottery combinations, you need to use the following formula (This is the tricky part):

Total number of possible combinations = n! ÷ ((n-r)! × r!)
N equals the amount of numbers to pick from, and R equals the amount of numbers you’ll need to choose

The exclamation mark in the formula is called a ‘factorial’ – it means multiply that number by every number that precedes it. For example, 5! = 5 × 4 × 3 × 2 × 1 =120.

So, to use the Health Lottery as an example, you choose 5 numbers from a possible 50. This means that:

The total number of possible combinations = 50!  ÷ ((50 – 5)! × 5!) = 2118760

How Many Lottery Combinations are there in the UK’s National Lottery?

There are 45,057,474 different combinations possible in the National Lottery. You get this by using the same formula from above, but you choose 6 numbers from a possible 59:

The total number of possible combinations = 59!  ÷ ((59 – 6)! × 6!) = 45,057,474

This is a much larger number of combinations than there are in The Health Lottery – which is why the odds of winning the Health Lottery are so much better!

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