Decision Tree - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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CS 173:

Discrete Mathematical Structures

Decision Tree

Suppose you have 4 shirts, 3 pairs of pants,

and 2 pairs of shoes. How many different

outfits do you have?

Pigeonhole Principle

If n pigeons fly into k pigeonholes and k < n, then

some pigeonhole contains at least two pigeons.

CS

Pigeonhole Principle

If n pigeons fly into k pigeonholes and k < n, then

some pigeonhole contains at least two pigeons.

We can use this simple little fact to prove amazingly complex things.

Pigeonhole Principle

Six people go to a party. Either there is a group

of 3 who all know each other, or there is a

group of 3 who are all strangers.

Consider one person.

She either knows or doesn’t know each other person.

But there are 5 other people! So, she knows, or doesn’t know, at least 3 others.

Let’s say she knows 3 others. If any of those 3 know each other, we have a blue , which means 3 people know each other. So they all must be strangers.

But then we’ve proven our conjecture for this case.

The case where she doesn’t know 3 others is similar.

Ice Cream Cones

Do these two cones provide the same ice cream

experience?

Permutations

Suppose you have time to listen to 10 songs on your

daily jog around campus. There are 6 Cake

tunes, 8 Moby tunes, and 3 Eagles tunes to

choose from.

How many different jog playlists can you make?

P(17,10) = 17x16x15x14x13x12x

Permutations

Suppose you have time to listen to 10 songs on your

daily jog around campus. There are 6 Cake

tunes, 8 Moby tunes, and 3 Eagles tunes to

choose from.

Now suppose you want to listen to 4 Cake, 4 Moby,

and 2 Eagles tunes, in that band order. How

many playlists can you make?

P(6,4) x P(8,4) x P(3,2)

Permutations

In how many ways can 5 distinct Martians and 3

distinct Jovians stand in line, if no two Jovians

stand together?

M1 M2 M3 M4 M

5! X P(6,3)

Combinations

A combination is an unordered selection of

elements from some set.

The number of combinations of r distinct objects

chosen from n distinct objects is denoted by

C(n,r) or nCr or n , and is read “n choose r.”

r

 



 



C(n,r) = P(n,r)/r! = n!/((n-r)!r!)

Combinations

A committee of 8 students is to be selected from

a class consisting of 19 frosh, and 34 soph.

In how many ways can 3 frosh and 5 soph be

selected?

Combinations

A committee of 8 students is to be selected from

a class consisting of 19 frosh, and 34 soph.

In how many ways can a committee with exactly 1

frosh be selected?

Combinations

A committee of 8 students is to be selected from

a class consisting of 19 frosh, and 34 soph.

In how many ways can a committee with at least 1

frosh be selected?