Permutations and Combinations

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:

nr

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

Combination 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)

nr = 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)

Permutation Example

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) = nPr = nPr = Permutation Without Repetition

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:

Combination With Repetition

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:

Combination 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 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)

Combination Example 1

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 Example 2

Combination Notation

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

C(n,r) = nCr = nCr = Combination Notation

C(7,4) =35

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