When does order matter?

When the positions are distinguishable. A podium of gold, silver and bronze is a permutation; a committee of three is a combination. Ask yourself whether swapping two chosen items produces a different outcome.

Why can I not just use factorials directly?

Because 171! overflows what a computer can hold as a number, while 200 choose 3 is only 1,313,400. This page multiplies and divides in step so the intermediate values stay small, which is why large draws still work here.

What if items can repeat?

These formulas assume each item is taken at most once. Drawing with replacement — where the same number can come up twice — uses different arithmetic and gives larger counts.

Math

Combination Calculator

Enter how many items there are and how many you take to see the combinations and permutations.

The link carries your figures, so it reopens on exactly these numbers.

How fast order starts to matter

From the same pool you entered, drawing different numbers of items. The two columns differ by r factorial — the number of ways one chosen set can be arranged — and that gap grows far faster than either column.

Items drawn	Combinations (order ignored)	Permutations (order matters)	Orderings of the chosen set
2Order doubles the count. Still easy to hold in your head.	1,176	2,352	2
3	18,424	110,544	6
4Order now multiplies by 24.	211,876	5,085,024	24
5	1,906,884	228,826,080	120
6By six items, order multiplies the count by 720.	13,983,816	10,068,347,520	720

The formula

combinations = n! ÷ (r! × (n − r)!)

permutations = n! ÷ (n − r)!

permutations with repetition = n^r

combinations with repetition = (r + n − 1)! ÷ (r! × (n − 1)!)

orderings = r!

Worked example

The 6-from-49 lottery draw

49 choose 6 = 13,983,816 combinations
One ticket is one of them
1 ÷ 13,983,816 = 0.0000072% per draw

Where this goes wrong

Counting orderings the problem does not care about

Choosing 6 numbers from 49 gives 13,983,816 combinations but 10,068,347,520 permutations — 720 times more, because a lottery ticket does not care what order the balls come out. Using the ordered count makes a one-in-fourteen-million chance look like one in ten billion.

Questions

When does order matter?: When the positions are distinguishable. A podium of gold, silver and bronze is a permutation; a committee of three is a combination. Ask yourself whether swapping two chosen items produces a different outcome.
Why can I not just use factorials directly?: Because 171! overflows what a computer can hold as a number, while 200 choose 3 is only 1,313,400. This page multiplies and divides in step so the intermediate values stay small, which is why large draws still work here.
What if items can repeat?: These formulas assume each item is taken at most once. Drawing with replacement — where the same number can come up twice — uses different arithmetic and gives larger counts.