what is different between combination and permutation?

[goldEli]

reference

• when the order does matter, it is a permutation.
• when the order does not matter, it is a combination.

Permutation

1. Repetition is allowed:

Such as a permutation lock, the number is '333'.

For example, choose 3 of those things(n), the permutations are:

n*n*n

2. No repetition:

For example the three first people in a running race, you can't be first and second, assume there are five people, the permutations are:

5*4*3

!5 / !(5-3)

choose r from n:
!n / !(n-r)

Combination

1. Repetition is allowed

There are five flavors of ice cream: banana, chocolate, lemon, strawberry, vanilla.

We have three scoops, how many variations will there be?

let's use letters for the flavors: {b, c, l, s, v}. Examples selections include:

1. {c,c,c} {three scoops chocolate}
 >ooo>>>
2. {b,l,v} (one each of banana, lemon, vanilla)
 o>>o>>o
3. {b,v,v} (one scoop of banana, two scoop of vanilla)
 o>>>>o

!n / !r * !(n - r)

!(3 + 5 - 1) / !3 * !(5 - 1)

!(r + n - 1) / !r * !((r + n - 1) - r) = !(r + n - 1) / !r * !(n - 1)

2.No repetition

choose 3 from 3

the permutations are:
123
132
213
231
312
321

the combinations are:
123

3! / !(3-3)*!3

choose r from n:

n! / !(n-r)*!r