Permutations of Non-Distinguishable

Objects ... like twins. ;-)

My Lovely Twins

but for real this time ...

Permutations (the "Pick" Formula)

In how many ways can 5 people be seated in a straight line?

In how many ways can six students be seated in 8 vacant seats?

A permutation is an ordered arrangement of objects.

n is the number of objects available to be arranged

Examples:

On the calculator ... Press: [MATH]

[<] (Prb) [2] (nPr)r is the number of objects

that are being arranged.

(a) How many “words” of 4 different letters each can be made from the letters A, E, I, O, R, S, T?

(c) In how many of these words do vowels and consonants alternate?

(b) How many of these words begin with a vowel and end with a consonant?

HOMEWORK

(a) How many numbers of 5 different digits each can be formed from the digits 0, 1, 2, 3, 4, 5, 6?

(b) How many of these numbers are even?

HOMEWORK

(c) What is the probability that one of these numbers is even if the digits are randomly chosen ?

(a) In how many ways can 4 English books and 3 French books be arranged in a row on a shelf?

(b) In how many of these ways will the French books be together?

HOMEWORK

(c) What is the probability the French books will be together?

If a fair coin is tossed 4 times, what is the probability of obtaining exactly 2 heads?

T

TT

TT

T

Permutations of non-distinguishable objects ...

Find the number of different "words" that can be made by rearranging the letters in the word:

Examples:

(a) BOOK (b) MISSISSIPPI

The number of ways to arrange n objects that contain k , k , k , ... sets of non-distinguishable objects is given by:

321

If a fair coin is tossed 4 times, what is the probability of obtaining exactly 2 heads?

If a fair coin is tossed 4 times, what is the probability of obtaining exactly 2 heads?

All the letters of the word MANITOBA are arranged at random in a row. How many ways can this be done?

What is the probability that this random arrangement will have the two A’s next to each other?

How many arrangements will have the two A’s next to each other?

HOMEWORK

(a) In how many ways can the letters of the word GEOMETRY be arranged so that vowels and consonants alternate?

(b) In how many of these ways is Y the last letter?

(c) If one of these "words" is randomly selected, what is the probability that Y is the last letter?

HOMEWORK