How Lottery Combinations Are Calculated: The Math Behind 5/90

Last updated: August 2026

Lottery odds get quoted constantly and explained rarely. A player is told the chance of hitting all five numbers in a 5/90 game is roughly one in 44 million and simply has to take it on trust. The calculation behind that figure is not complicated, and working through it once tells you more about lottery than any amount of number analysis.

Combinations Versus Permutations

Everything starts with one distinction. A permutation counts arrangements where order matters. A combination counts selections where order does not.

Lottery draws are combinations. If the drawn balls are 4, 19, 33, 57 and 80, a ticket carrying those five numbers wins regardless of the sequence they came out in. Writing them as 80, 57, 33, 19, 4 makes no difference to the outcome. This matters because the number of arrangements is vastly larger than the number of selections, and confusing the two throws the odds out by an enormous factor.

Building the 5/90 Figure Step by Step

Take a game where five balls are drawn from a pool of ninety, without replacement, meaning a ball once drawn cannot appear again in the same draw.

Start by counting arrangements. The first ball can be any of 90. Once it is gone, the second can be any of the remaining 89, then 88, then 87, then 86. Multiplying those together gives the number of ordered sequences:

• 90 × 89 × 88 × 87 × 86 = 5,273,912,160

That figure counts every ordering separately, which overcounts heavily for our purposes. Any given set of five numbers can be arranged in 5 × 4 × 3 × 2 × 1 = 120 different orders, and all 120 are the same ticket.

So divide out the duplication:

• 5,273,912,160 ÷ 120 = 43,949,268

There are 43,949,268 distinct five-number combinations in a 90-ball pool. One ticket covers exactly one of them, which is where the familiar one in roughly 44 million comes from.

Why Smaller Bet Types Have Much Better Odds

Nigerian lotto rarely asks players to hit all five. Most staking is done on smaller targets, and the same method shows why those are so much more achievable.

Consider a two-number bet, where you pick two numbers and need both to appear among the five drawn. The number of two-number combinations available in a 90-ball pool is:

• (90 × 89) ÷ (2 × 1) = 4,005

Of those 4,005 pairs, the ones that win are the pairs that can be formed from the five drawn balls. That count is:

• (5 × 4) ÷ (2 × 1) = 10

So the probability of a winning pair is 10 divided by 4,005, roughly one in 400. Set that beside one in 43,949,268 and the gap is stark. Going from needing two numbers to needing five makes the outcome around 110,000 times harder to achieve.

The Rule Underneath All of It

The general formula for choosing r items from n, written as C(n, r), is n factorial divided by r factorial multiplied by (n minus r) factorial. In practice you rarely need the full expression. The shortcut used above works for any lottery bet type:

• Multiply n, then n minus 1, and so on for r terms in total
• Divide by r factorial, meaning r × (r−1) × … × 1

For a three-number bet from 90, that gives (90 × 89 × 88) ÷ 6 = 117,480 possible triples, with (5 × 4 × 3) ÷ 6 = 10 of them winning. The probability lands at roughly one in 11,748.

Where Permutation Betting Fits

Permutation betting, offered in various forms across Nigerian operators, is often misunderstood as improving your odds. It does not change the underlying probability of any single combination. What it does is let you cover many combinations at once from a larger set of selected numbers.

If you select eight numbers and the system stakes every possible pair from them, the number of pairs covered is (8 × 7) ÷ 2 = 28. You now hold 28 of the 4,005 possible pairs rather than one, so your chance of holding a winning pair rises to roughly 28 in 4,005, about one in 143.

The improvement is real, but it is bought and not conjured. You are staking 28 times, and the cost rises in step with the coverage. Each individual pair still carries its original one in 400 probability, and the house edge on each of those stakes is unchanged. Permutation is a convenience for placing many bets at once, not a method for beating the maths.

What the Numbers Are Actually Telling You

Running these calculations makes one thing unavoidable: the odds are fixed by the structure of the game itself, not by which numbers you write down. Every one of the 43,949,268 five-number combinations carries identical probability. The sequence 1, 2, 3, 4, 5 is exactly as likely as any scattered-looking set, because the machine has no concept of what looks random to a human eye.

This is also why no selection system can shift the odds. A system can change which combinations you hold and how many of them you hold, and the second of those genuinely does improve your chance because you have paid for more entries. Nothing available to a player changes the probability attached to a single entry. Understanding that distinction is the difference between staking with clear eyes and staking on a story.