## Permutations and Combinations

Table of Contents

**Definitions**

- a Combination is when the order doesn’t matter.
- a Permutation is When the order does matter.

In other words:

A Permutation is an ordered Combination.

**Permutations**

There are 2 types of permutation:

- Permutation with Repetition: such as the lock. It could be “444”.
- Permutation without Repetition: for example the first three people in a running race. You can’t be first and second.

**Permutation with Repetition**

The formula is written:

*n*^{r}

where,

- n is number of things to choose from
- r is number of things we choose of n
- repetition is allowed
- order matters

**Permutation without Repetition**

where,

- n is number of things to choose from
- r is number of things we choose of n
- repetition is
**NOT**allowed - order matters

**Permutation Examples**

1. For my account pin, there are 10 numbers to choose from (0,1,2,3,4,5,6,7,8,9) and we choose 6 of them. (You can choose same number twice or more)

n^{r }= 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000

So there are 1,000,000 permutations for my pin.

2. What order could 7 runners be in top 3? (the runner in first place can’t be in second place)

So there are 210 permutations for top 3 runners.

**Permutation Notation**

Instead of writing the whole formula, people use different notations such as these:

*P*(*n,r*) = * ^{n}P_{r}* =

*=*

_{n}P_{r}*P*(7,3) = 210

**Combinations**

There are 2 types of combinations (remember the order does not matter now):

- a combination with Repetition: such as coins in your pocket (5,5,10,10,10)
- a combination without Repetition: such as lottery numbers (5,14,17,22,30,34)

**Combination with repetition**

The formula is written:

where,

- n is number of things to choose from
- r is number of things we choose of n
- repetition is allowed
- order doesn’t matters

**Combination without repetition**

The formula is written:

where,

- n is number of things to choose from
- r is number of things we choose of n
- repetition is
**NOT**allowed - order doesn’t matters

**Combination Examples**

1. There are strawberries, grapes, bananas, pineapples, and apples in refrigerator. I Want to make 3 cups of juice. How many combinations i can make? (repetition is allowed)

So, there are 420 combinations of juices i can make.

2. My teacher tell me to choose 3 girls from 7 girls in my class. How many choices i have? (repetition is not allowed)

**Combination Notation**

Instead of writing the whole formula, people use different notations such as these:

*C*(*n,r*) = * ^{n}C_{r }*=

*=*

_{n}C_{r}*C*(7,4) =35