probability
did you know six numbers can hide millions of combinations?
a tiny-looking ticket sits inside a huge mathematical space.
13,983,816
different sets of six
choose six distinct numbers from 49, ignoring their order.
- 49 available
- choose 6
- one exact set
the idea
in a hypothetical draw of six different numbers from 49, there are 13,983,816 possible sets. one specific set has exactly the same chance as any other, including 1, 2, 3, 4, 5, 6.
how it works
a set of six numbers can be written in 720 different orders. since the draw ignores order, divide the ordered count by 720. the result counts each possible ticket exactly once.
go deeper
why not just multiply 49 by six?
in a hypothetical draw of six different numbers from 49, there are 13,983,816 possible sets. one specific set has exactly the same chance as any other, including 1, 2, 3, 4, 5, 6.
each choice changes what remains. there are 49 options for the first number, then 48, then 47. multiplying counts the ways to make an ordered list, which is more than we want.
remove the duplicate orders
a set of six numbers can be written in 720 different orders. since the draw ignores order, divide the ordered count by 720. the result counts each possible ticket exactly once.
ten different tickets cover ten of those possible sets. that multiplies the chance by ten, but the absolute chance remains very small: about one in 1.4 million for this simplified match-all game.
what does the choose symbol mean?
49 choose 6 counts unordered selections without replacement. the factorial notation is just a compact way to write the descending products.
C(49, 6) = (49 × 48 × 47 × 46 × 45 × 44) / (6 × 5 × 4 × 3 × 2 × 1) = 13,983,816
why do patterns not change the odds?
- the drawing mechanism does not label some sets as random-looking. every specific set has the same probability in a fair draw.
- real games can add bonus balls and different prize tiers. those rules need their own calculations; this is an invented six-from-49 example.
sources & further reading
concepts: combinations · factorials · sample spaces
2 minute read