Some lovely and slightly random facts…

Give the number of objects described.

1. The number of cards in a standard deck 52522. The number of cards of each suit in a standard deck 13133. The number of faces on a cubical die 664. The number of possible totals when two dice are rolled 11115. The number of vertices of a decagon 10106. The number of musicians in a string quartet 44

More lovely and slightly random facts…

Give the number of objects described.

7. The number of players on a soccer team 11118. The number of prime numbers between 1 and 10, inclusive449. The number of squares on a chessboard 6464

10. The number of cards in a contract bridge hand 1313

11. The number of players on a rugby team 1515

Chapter Chapter 9:1a 9:1a

Discrete Discrete MathematiMathemati

cscs

What is Discrete What is Discrete Mathematics???Mathematics???

Any interval (Any interval (aa, , bb) contains a ) contains a continuumcontinuum of real of realnumbers numbers you can “zoom in” forever and there will you can “zoom in” forever and there willstill be an interval there…still be an interval there…

In contrast, In contrast, discrete mathematicsdiscrete mathematics involves separate involves separate(or “discrete”) numbers that (or “discrete”) numbers that do notdo not lie on a continuum. lie on a continuum.The simplest type of discrete mathematics is…The simplest type of discrete mathematics is…

The mathematics of the continuum is used a greatThe mathematics of the continuum is used a greatdeal in algebra, analysis, and geometry, which alldeal in algebra, analysis, and geometry, which alllead to important concepts in calculus…lead to important concepts in calculus…

COUNTING!!!COUNTING!!!

The Importance of CountingIn how many ways can three distinguishable objects bearranged in order? (let’s call them A, B, and C)

We can “see” the solution to this problem with atree diagram:

StartingPoint

A

B

C

B CC BA

CC

AA

BB

A

The six differentorderings:

ABC, ACB,BAC, BCA,CAB, CBA

Multiplication Principle of Counting

(our tree diagram gives a visualization of this principle…)

If a procedure P has a sequence of stages S ,S , … , S and if

1

2 n

S can occur in r ways,1 1

S can occur in r ways,2 2

S can occur in r ways,n n

then the number of ways that the procedure P canoccur is the product: r r r .1 2 n

Multiplication Principle of Counting

Important note about this principle!!!Important note about this principle!!!

Be mindful of how the choices atBe mindful of how the choices ateach stage are affected by theeach stage are affected by thechoices at preceding stages…choices at preceding stages…

(think about how this applies(think about how this appliesIn our very first example)In our very first example)

Using the Multiplication PrincipleA certain state’s license plates consist of three lettersfollowed by three numerical digits (0 through 9). Find thenumber of different license plates that could be formed

(a) if there is no restriction on the letters or digits;

(b) if no letter or digit can be repeated.

With no restrictions, we have 26 ways to fill each of thefirst three blanks, and 10 ways to fill each of the finalthree blanks.

Multiplication Principle: 26 x 26 x 26 x 10 x 10 x 10

= 17,576,000 ways

Using the Multiplication Principle

A certain state’s license plates consist of three lettersfollowed by three numerical digits (0 through 9). Find thenumber of different license plates that could be formed

(a) if there is no restriction on the letters or digits;

(b) if no letter or digit can be repeated.

If no repeats are allowed, there are 26 choices for the firstblank, 25 for the second, 24 for the third, 10 for the fourth,9 for the fifth, and 8 for the sixth.

Multiplication Principle: 26 x 25 x 24 x 10 x 9 x 8

= 11,232,000 ways

Are these realistic estimates for #’s of ways???Are these realistic estimates for #’s of ways???

Permutations

What are they???

Permutations – possible orderings of a set of objects

Consider the There are 24 possibleorderings………………………can we generalize for a rule ?

RULE: If there are n objects in a set, then thereare n ! permutations of the setNote: the symbol n! (read “n factorial”) represents theproduct n(n – 1)(n – 2)(n – 3)…(2)(1). In addition, wedefine 0! = 1.

What are they???

Permutations – possible orderings of a set of objects

RULE: If there are n objects in a set, then thereare n ! permutations of the set

This rule only works if all elements of a set aredistinguishable from each other…

Distinguishable Permutations

There are n! distinguishable permutations of an n-set containingn distinguishable objects.

If an n-set contains n objects of a first kind, n objects of asecond kind, and so on, with n + n + … + n = n, then thenumber of distinguishable permutations of the n-set is

1 2

1 2 k

1 2 3

!

! ! ! !k

n

n n n n

Some Practice ProblemsCount the number of different 9-letter “words” (don’t worryabout whether they’re in the dictionary) that can be formedusing the letters in each of the given words.

1. DRAGONFLY

Each permutation of the 9 letters forms a different word.

There are 9! = 362,880 such permutations.

2. BUTTERFLY

There are also 9! permutations, but simply switching thetwo T’s does not result in a new word.

There are = 181,440 distinguishable permutations.9!

2!

Some Practice ProblemsCount the number of different 9-letter “words” (don’t worryabout whether they’re in the dictionary) that can be formedusing the letters in each of the given words.

3. BUMBLEBEE

The three B’s and three E’s are indistinguishable, so wedivide by 3! twice to correct for the overcount…

9!

3!3!There are = 10,080 distinguishable permutations.

Permutation Counting Formula

In some counting problems, we are interested in using nobjects to fill r blanks in order, where r < n. These are calledpermutations of n objects taken r at a time.

The number of permutations of n objects taken r at a time isdenoted P and is given byrn

!

!n r

nP

n r

for 0 r n

If r > n, then 0n rP What is ?n nP

More Practice ProblemsEvaluate the given expressions, first without a calculator.Then, check your answer with a calculator.

6!

6 4 !

1.6 4P

6!

2!

6 5 4 3 2 1

2 1

360

11!

11 3 !

2.11 3P

11!

8! 11 10 9 990

More Practice ProblemsEvaluate the given expression.

!

3 !

n

n

3.

3nP

1 2 3 2 1

3 2 1

n n n n

n

1 2n n n

More Practice ProblemsSixteen actors answer a casting call to try out for roles asdwarfs in a production of Snow White and the Seven Dwarfs.In how many ways can the director cast the seven roles?

57,657,60016 7P

There are 7 blanks to be filled, and 16 actors with whichto fill them:

ways

More Practice ProblemsProctor has 22 players train for his rugby team, and he mustfill 15 playing spots for a weekend match. Assuming that allplayers are equally qualified for each playing position, in howmany ways can Proctor fill the spots?

172.23016 10 22 15P ways