14B - Permutations and probability

Permutations

Factorials

Factorial notation is used tor represent the multiplication of consecutive natural numbers (ℕ). For a natural number, n, we define:

• We read n! as 'n factorial'.
• We also define 0! = 1 (this will be important later).
• We can also separate a factorial out i.e. n! = n × (n - 1)!

14B - VIDEO EXAMPLE 1:

Evaluate the following factorials

Permutations of n objects

A permutation is an arrangement of a collection of objects where order is important.

• When arranging n different objects in a row, without restriction, there are n! ways.

14B - VIDEO EXAMPLE 2:

In how many ways can 7 students be arranged in a row?

Permutations of n objects taken r at a time

When we have n objects but only arrange r at a time we can denote this as: ⁿP_r

14B - VIDEO EXAMPLE 3:

In the Olympics, the final of the 100 meter freestyle has 8 competitors. In how many ways can the gold, silver and bronze medals be awarded? Assume no two swimmers finish at the same time (known as a dead heat).

14B - VIDEO EXAMPLE 4:

Consider the numbers 0, 1, 2, 3, ..., 9.

• How 4-digit pass-codes exist if none of the numbers can be repeated?
• How 4-digit pass-codes exist if any number can be repeated?

Permutations with restrictions

When dealing with arrangements that involve restrictions, deal with the restrictions first.

Representation using boxes

We can uses boxes to represent the positions in our arrangement, this is useful for more complex situations that involve restrictions.

14B - VIDEO EXAMPLE 5:

A family are flying overseas. They plan to visit England, Scotland, Wales, France and Germany. Given the family land at Heathrow Airport in England and they fly out from Frankfurt Airport, Germany, in how many ways can the family’s journey be planned?

14B - VIDEO EXAMPLE 6:

Consider the numbers 0, 1, 2, 3, 4 and 5. If each digit can only be used once, determine:

• The amount of 4-digit numbers without restriction.
• The amount of 4-digit even numbers.
• The amount of even numbers less than four thousand.

Permutations involving groups

For some arrangements, we may want (or need) certain objects to be grouped together. When dealing with permutations involving groups:

1. Treat the group as a single item and arrange all items including the grouped item.
2. Next, multiply by the number of possible arrangements in the grouped item.

14B - VIDEO EXAMPLE 7:

Matt has booked an around the world trip. He plans to visit Australia, Africa, South America, North America, Europe and Asia. In how many ways can he plan his trip if he decides to visit both of the America’s successively?

Arrangements involving identical objects

In some situations identical objects may be being arranged; however, when you swap identical objects with each other no new (unique) arrangement is created. Therefore, when arranging n objects, where n₁ are alike, n₂ are alike ... and n_r are alike, the number of permutations is given by:

14B - VIDEO EXAMPLE 8:

A mathematics teacher has two identical Specialist Maths textbooks, three identical Methods textbooks and six identical Further Maths textbooks to arrange on a shelf. In how many ways can this be done?

Applications of permutations to probability

The following examples illustrate the application of permutations to probability. Further examples involving combinations can be found here.

14B - VIDEO EXAMPLE 9:

A deck of cards is randomly shuffled and 2 cards are drawn without replacement. What is the probability two Kings will be drawn?

14B - VIDEO EXAMPLE 10:

In a family of five, Jess and Maddy are twins that do not like to sit next to each other. On a flight from Melbourne to Sydney the family book 5 seats in a row. What is the probability that Maddy and Jess will not be allocated seats next to each other?