Permutations and Combinations

Solve Counting Problems Using Multiplication PrincipleSolve Counting Problems Using PermutationsSolve Counting Problems Using Combinations

Solve Counting Problems Using Permutations involving non distinct objects

Multiplication Principle of Counting

If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice and r selections for the third choice and so on then the task of making these selections can be done in

p∙q∙r∙……..

Question

The fixed price dinner at a restaurant provides the following choices

Appetizer: Soup or SaladEntrée: Chicken, Beef, Fish or PorkDessert: Ice Cream or CheesecakeHow many different meals can be ordered?

Permutation

A permutation is an ordered arrangement of r objects chosen from n objects

Permutations: Distinct objects with repetition

The number of ordered arrangements of r objects chosen from n objects in which the n objects are distinct and repetition is allowed is equal to nr.

The symbol P(n,r) represents the number of ordered arrangements of r objects chosen from n distinct objects where r≤ n and repetition is not allowed.

Permutations of r objects chosen from n distinct objects without repetition

The number of arrangements of n objects using r ≤ n of them in which

1. The n objects are distinct

2. Once an object is used it can not be used again

3. Order is important

Is given by the formula

)!(

!),(

rn

nrnP

Example

The International Airline Transportation Association assigns three letter codes to represent airport locations. For example the airport code for Ft Lauderdale, Florida is FLL. Notice that repetition is allowed in forming this code. How many airport codes are possible?

Example

Suppose that we wish to establish a three letter code using any of the 26 uppercase letters of the alphabet, but we require that no letter be used more than once. How many different three letter codes are there?

Lining up people

In how many ways can 5 people be lined up?

Birthday problem

All we know about Shannon, Patrick and Ryan is that they have different birthdays. If we listed all the possible ways this could occur how many would there be? Assume there are 365 days in a year

Combinations

A combination is an arrangement, without regard to order, of r objects selected from n distinct objects without repetition, where r ≤n. The symbol C(n,r) represents the number of combinations of n distinct objects using r of them.

Forming Committees

How many different committees of 3 people can be formed from a pool of 7 people?

Forming Committees 2

In how many ways can a committee consisting of 2 faculty members and 3 students be formed if 6 faculty members and 10 students are eligible to serve on the committee?

Forming different words

How many different words (real or imaginary) can be formed using all the letters in the word REARRANGE?

Number of combinations of n distinct objects taken r at a time

The number of arrangements of n objects using r≤n of them, in which

1. Th n objects are distinct

2. Once an object is used, it cannot be repeated

3. Order is not important

is given by the formula

!)!(

!),(

rrn

nrnC

Permutations involving n objects that are not distinct

The number of permutations of n objects of which n1 are of one kind and n2 are of a second kind…….and nk are of the kth kind is given by

!...!!

!

21 knnn

n

Question

The student relations committee of a college consists of 2 administrators 3 faculty members and 5 students. Four administrators, 8 faculty members, and 20 students are eligible to serve. How many different committees are possible?

Question

How many different 9 letter words (real and imaginary) can be formed from the letters in the word ECONOMICS?

Question

How many different 11 letter words (real or imaginary) can be formed from the letters in the word MATHEMATICS?