Combinations, arrangements and permutations calculator

The key difference is whether order matters.

Combination (order does NOT matter)

Choosing a group regardless of order — like lottery numbers.

C(n,r) = n! ÷ [r!·(n−r)!]

Arrangement (order matters)

Choosing and ordering — like a podium of 1st, 2nd and 3rd.

P(n,r) = n! ÷ (n−r)!

Step-by-step example

From 5 people, pick 2:

• Combination (a pair): C(5,2) = 10
• Arrangement (1st and 2nd): P(5,2) = 20

The arrangement gives double because "Ana-Bruno" and "Bruno-Ana" count as different.

The question that decides which to use

Just one: does reordering create a different outcome? If yes, it is an arrangement. If no, it is a combination. A race podium changes with order; cards in a hand do not. A four-digit PIN changes with order; a four-person committee does not.

Why lotteries are so improbable

In a 6-from-49 draw the order does not matter, so it is a combination: C(49,6) = 13,983,816 possible tickets. A single line therefore has 1 chance in roughly 14 million.

It is worth seeing what buying more numbers does: covering 7 numbers gives C(7,6) = 7 lines, raising the chance to 1 in 2 million — seven times better, for seven times the price. The price-to-odds ratio does not change; only the absolute amount risked does.