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n Choose k: How 13,983,816 6/49 Combinations Are Counted
The counter above lists every line of a small drum and gives n choose k for any drum. Below: the combinations formula, a count by hand and the totals for Canadian games.
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The combinations formula
n choose k = n! ÷ (k! × (n − k)!), where n! = n × (n − 1) × … × 1
Example: Lotto 6/49: n is forty-nine and k is six, the two numbers in the game's name, so n choose k = 13,983,816 possible lines.
Counting a small drum
- Label the ballsTake a toy drum with balls A, B, C, D and E. Each line picks two of them.
- Start with ball APair A with every later ball: AB, AC, AD, AE.
- Move to ball BPair B only with later balls: BC, BD, BE. BA is the same line as AB, so skip it.
- Finish the listThen CD, CE and DE. Nothing new is left.
- Count and checkCount the lines in the list. Five choose two gives the same total. The counter above shows the list and the formula result.
Combinations in Canadian drums
| Game or draw | Drum (k from n) | Combinations |
|---|---|---|
| Lotto 6/49 | Six from forty-nine | 13,983,816 |
| Lotto Max, larger drum | Seven from fifty-two | 133,784,560 |
| Lotto Max, smaller drum | Seven from fifty | 99,884,400 |
| Daily Grand, main numbers only | Five from forty-nine | 1,906,884 |
| Keno, smaller board | Twenty drawn from seventy | 161,884,603,662,657,900 |
| Keno, larger board | Twenty drawn from eighty | 3,535,316,142,212,174,300 |
Each combination has the same chance. Quick pick and your own numbers give the same odds. The game conditions set the drum or board size of each game.
Combinations vs prize odds
Combinations
- Count every possible line in one drum
- One number for each drum, from n choose k
- Stay the same while the drum stays the same
Prize odds
- In Lotto 6/49, one combination matches all six numbers for the jackpot
- Smaller prizes count lines that match only some numbers
- See any game, any prize
Words on this page
- n
- How many balls are in the drum.
- k
- How many balls one line picks.
- Factorial (n!)
- n multiplied by every whole number below it, down to 1.
- Combination
- One set of k balls where order does not matter.
- Line
- One combination played on a ticket.
- Hypergeometric
- Counting how many lines match some numbers but not others. Used for keno odds.
Questions
How do you calculate n choose k?
Divide n! by k! and by (n − k)!. In practice, multiply k numbers counting down from n, then divide by k!. The counter above does this for any drum.
How many combinations are there in Lotto 6/49?
There are 13,983,816. Each one is a possible set of six numbers, and every set has the same chance. See Lotto 6/49 odds for each prize.
How many Lotto Max combinations are there?
Lotto Max picks seven numbers. With the larger drum the count is 133,784,560; with the smaller drum it is 99,884,400. The game conditions say which drum is used.
Does the order of numbers matter in a lottery combination?
No. The same numbers in any order are one line. That is why the combinations formula divides by k!.
Why are keno combination counts so large?
A keno draw picks many numbers from a big board, so the number of possible draws is huge: 161,884,603,662,657,900 on the smaller board. See keno math.