Section2.2_Introducing_Permutations_and_Factorial_Notation.notebook October 17, 2013

Section 2.2: Introducing Permutations and Factorial Notation

Factorial Notation is useful when calculating the number of permutations or combinations for specific problems.

Permutation • An arrangement of objects in a definite order, where each object appears only once in each arrangement. • How many different rearrangements can you have • Order is IMPORTANT • Two objects a and b have two permutations ab and ba. Combination • A grouping of objects where order is not important. • Two objects a and b have one combination because ab is the same as ba.

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Most bankcards can be used to access an account by entering a 4- digit personal identification number (PIN). However knowing these four digits is not enough to access the account. The digits have to be in the correct order.

Consider Leida‛s account, where four digits used in opening it are 1, 3, 7, and 9. Leida knows the first digit used is 1 but cannot remember the order of the remaining digits. We can arrange these remaining digits in six different ways.

379 397 739 793 937 973

The order is important, so we must consider each of these as a different arrangement. There are six possible arrangements to consider and only one of these arrangements will access the account. These types of arrangements are called PERMUTATIONS.

Permutation: an arrangement of a set of objects

If the order were not important, then in this case it would suggest that all six of these arrangements would access the account.

Combination: a selection from a group of objects without regard to order

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Ex. Determine the number of ways you can arrange 5 people in a line.

This product can be written in a compact form as 5! read as 5 factorial.

Factorial Notation • a concise representation of the product of consecutive descending natural numbers

n! = n(n −1)(n − 2)(n −3)...(2)(1)

For example: 5! = 5 × 4 × 3 × 2 ×1 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 (n+3)! = (n + 3) × (n + 2) × ( n +1 ) × (n) × ( n ­ 1)× … 3 ×2 × 1

Note: n! = n(n −1)! 5! = 5 × 4! 9! = 9 × 8!

Note (­2)! and (1/2)! have no meaning since ­2 and 1/2 are not natural numbers.

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Evaluating Factorials

• To evaluate a factorial means to find out its value. • Your answer will be a number • For some you can use a calculator, but others require mental math

Examples

1. Evaluate.

a) 8!

b) 5 8!

c)

d)

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e)

f)

2. Write the following expressions using factorial notation.

a)

b) 7 x 6 x 5

c) 10 x 9 x 8 x 7 x 6

d)

e)

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Simplifying Factorials

• To simplify a factorial expression > Look for cancellations on the top and bottom of fractions > Look to combine terms • Your answer will be a simpler expression, usually without a factorial

Remember n! = n(n −1)(n − 2)(n −3)...(2)(1) and n must be a natural number (ie. it must be greater than 0)

1. Simplify.

a)

b)

c)

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d)

e)

f)

g)

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Solving Factorial Equations

To solve a factorial expression: • Look for cancellations that will eliminate the factorial • Solve using algebra > Factor or quadratic formula • Check your solutions by substituting them into the original (anything that produces a negative factorial must be rejected) • Your answer will be a number

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1. Solve for n where n ∈ I.

A)

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B)

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C)

D)

E)

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Permutation • An arrangement of objects in a definite order, where each object appears only once in each arrangement. For example, the set of 3 objects a, b, c, can be listed in 6 different ordered arrangements or permutations.

Grad Committee: 3 people are running for the positions of president, vice­president, and treasurer. How many different ways can this be done?

The number of permutations from a set of n different objects, where n of them are used in each arrangement, can be calculated using n!.

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1. A six­digit secret code number uses the digits 1 through 6 exactly once each. If the first digit of the code number is 3 and the last digit is 2, how many possibilities are there for the code number?

2. a) How many arrangements are possible using all of the letters in WHISTLER?

b) How many arrangements of the letters in WHISTLER end with a W?

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3. Visitors to a movie website will be asked to rank 28 different movies. The website will present the movies in a different order for each visitor to reduce bias in the poll. How many permutations of the movie list are possible?

Do questions 7 ­ 9, 12 ­ 15 page 82 ­ 83.

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