Lottery Combinations Math, Explained Step by Step

Order matters, or it doesn't

In a Pick 4 game, 1-2-3-4 and 4-3-2-1 are different straight bets, so order matters. In a pick-5 game like a 5-of-39 draw, the order doesn't matter: 3, 11, 20, 27, 38 is the same set no matter how you write it.

Step 1: count ordered picks

There are 39 choices for the first ball, 38 for the second, then 37, 36 and 35. Multiply them: 39 × 38 × 37 × 36 × 35 = 69,090,840 ordered sequences.

Step 2: remove repeated orders

Any set of 5 numbers can be arranged 5 × 4 × 3 × 2 × 1 = 120 ways. Since order doesn't matter, divide: 69,090,840 ÷ 120 = 575,757.

The shortcut formula

Mathematicians write this as "n choose r", or C(n, r) = n! ÷ (r! × (n − r)!). Most calculators have an nCr button.

Why it matters

Knowing where odds come from helps you set expectations. Apollo AI's Recommendation Generator picks one of these combinations for you, and it's exactly as likely as any other.

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General information for US readers, not legal or financial advice. Lottery draws are random.