How Are Lottery Odds Calculated? Combinations, nCr and the 6/49 Example

The number 13,983,816 appears in every article about Lotto, but few show where it comes from. It is one of the simplest examples of combinatorics from school maths: understand what a combination is and compute one fraction. You can check every step in the combinations calculator or the lottery odds calculator.

What a combination is

A combination is a choice of several items from a set where the order does not count. Take a small example: five balls numbered 1, 2, 3, 4 and 5, from which we choose three. All possible choices are:

1-2-3, 1-2-4, 1-2-5, 1-3-4, 1-3-5, 1-4-5, 2-3-4, 2-3-5, 2-4-5, 3-4-5.

There are ten. The set 3-1-2 is not a new choice — it is the same three balls as 1-2-3, only written in a different order. The number of combinations of k items out of n is written C(n, k), “n choose k” or nCr. Here C(5, 3) = 10.

Why the order of numbers does not matter

A Lotto ticket is a set of numbers, not a sequence. The machine releases the balls in random order, but a win is checked only by whether your numbers are among the drawn ones. That is why we count combinations, not arrangements.

The five-ball example shows it. If order mattered, we would choose the first ball in 5 ways, the second in 4 and the third in 3 — 5 × 4 × 3 = 60 sequences in total. But every set of three balls can be arranged in 3! = 3 × 2 × 1 = 6 ways, so each set appears six times among those 60 sequences. The number of combinations is 60 ÷ 6 = 10 — exactly as many as we listed above.

The binomial coefficient formula

The same reasoning in general form gives the formula:

C(n, k) = n! ÷ (k! × (n − k)!)

The exclamation mark is the factorial: n! is the product of all whole numbers from 1 to n, e.g. 4! = 1 × 2 × 3 × 4 = 24. The top counts all arrangements and dividing by k! removes the different orders of the same set. With large numbers the factorials are huge, so in practice there is a shorter way — shown in the example below.

Calculating C(49, 6) step by step

Method one — through sequences, as with the five balls:

1. You pick the first number in 49 ways, the second in 48, and so on: 49 × 48 × 47 × 46 × 45 × 44 = 10,068,347,520 sequences.
2. Every set of six numbers can be arranged in 6! = 720 ways.
3. Number of sets: 10,068,347,520 ÷ 720 = 13,983,816.

Method two — multiply and divide in turn. The result after every step is a whole number, so it is easy to check on a calculator:

Step	Operation	Result
k = 1	49 ÷ 1	49
k = 2	49 × 48 ÷ 2	1,176
k = 3	1,176 × 47 ÷ 3	18,424
k = 4	18,424 × 46 ÷ 4	211,876
k = 5	211,876 × 45 ÷ 5	1,906,884
k = 6	1,906,884 × 44 ÷ 6	13,983,816

C(49, k) for consecutive k — each row is the previous one multiplied and divided

You get a bonus on the way: C(49, 5) = 1,906,884 is the number of all possible sets of five out of 49 numbers.

Probability versus odds of “1 in X”

All sets are equally likely and exactly one wins. The probability of the jackpot is therefore:

P = 1 ÷ 13,983,816 ≈ 0.0000000715 = 0.00000715%

“1 in 13,983,816” is the reciprocal of that probability: it says how many equally possible cases contain one favourable case. In everyday language both mean the same. In strict betting language “odds” are counted differently — as unfavourable cases to favourable ones, i.e. 13,983,815 to 1 — but with numbers this large a difference of one changes nothing.

A common mistake: 0.0000000715 and 0.00000715% are the same number. A percentage is simply the probability multiplied by 100.

Odds of three, four and five matches

For the lower tiers we count in how many ways you can match exactly k of the 6 drawn numbers and at the same time 6 − k of the other 43. This is the hypergeometric distribution — explained in detail in Hypergeometric Distribution Explained.

Matches	Favourable sets	Odds
6	1	1 in 13,983,816
5	6 × 43 = 258	1 in 54,201
4	15 × 903 = 13,545	1 in 1,032
3	20 × 12,341 = 246,820	1 in 56.66

Matches in Lotto — favourable sets and odds

For three matches, for example: C(6, 3) = 20 ways to choose the three matched numbers and C(43, 3) = 12,341 ways to choose the three missed ones. Together 20 × 12,341 = 246,820 sets, and 13,983,816 ÷ 246,820 ≈ 56.66.

Example: several tickets and many draws

Ten different tickets for one draw cover ten different sets out of 13,983,816. The chance of the jackpot is then exactly 10 ÷ 13,983,816, or 1 in 1,398,381.6 — ten times higher and ten times more expensive. Each single ticket still has 1 chance in 13,983,816.

Playing over many draws works differently. One ticket in each of 156 draws in a year gives a chance of at least one jackpot of 1 − (1 − p)¹⁵⁶, where p = 1 ÷ 13,983,816. That is about 0.0011%, roughly 1 in 89,640. The result is almost exactly 156 × p, because with such a small p the chances simply add up. You can calculate both cases in the multiple tickets calculator, and the cost of a full system in the lottery system calculator.

Exercises

Level: secondary school. Try to work it out yourself first, then open the solution.

Exercise 1. Mini Lotto

How many different bets are possible in a 5/42 game?

Show solution

1. Count combinations: C(42, 5) = 42 × 41 × 40 × 39 × 38 ÷ 5!.
2. Top: 42 × 41 × 40 × 39 × 38 = 102,080,160.
3. 5! = 120, so 102,080,160 ÷ 120 = 850,668.

Answer: 850,668 bets, so the odds of five matches are 1 in 850,668.

Check in the combinations calculator

Exercise 2. Three tickets

You buy three different Lotto tickets for the same draw. What is the probability of the jackpot?

Show solution

1. Three different tickets are three different sets out of 13,983,816.
2. The events “ticket 1 wins”, “ticket 2 wins” and “ticket 3 wins” exclude each other, so the probabilities add up.
3. P = 3 ÷ 13,983,816 = 1 ÷ 4,661,272.

Answer: 1 in 4,661,272 (about 0.0000215%).

Compare with the multiple tickets calculator

Exercise 3. A system of 8 and of 10

How many times more bets does a full system of 10 numbers cover than a system of 8 numbers in Lotto?

Show solution

1. System of 8 numbers: C(8, 6) = 28 bets.
2. System of 10 numbers: C(10, 6) = 210 bets.
3. 210 ÷ 28 = 7.5.

Answer: 7.5 times more bets — and a 7.5 times higher cost.

Work out the cost in the system calculator

Exercise 4. From “1 in X” to a percentage

Convert odds of 1 in 850,668 into a percentage.

Show solution

1. The probability is 1 ÷ 850,668 ≈ 0.0000011755.
2. Percentage: 0.0000011755 × 100 = 0.00011755%.

Answer: about 0.000118%.

See it in the lottery odds calculator

Calculate it yourself

• Combinations, variations and permutations calculator — C(n, k) with every step.
• Lottery Odds Calculator — any number of matches in Lotto, Eurojackpot or your own lottery.
• Multiple Tickets Lottery Odds — what more tickets and more draws really change.
• Probability Academy — all calculators and articles in one place.