Generating Function pptx, Slides of Mathematics

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COUNTING ARGUMENTS IN

ALGEBRA

Alvin Ling

TAS Math Team

WARMUP

Warmup 1. Real numbers and are chosen independently

and uniformly at random from the interval. What is the

probability that , where denotes the greatest integer less than

or equal to the real number?

Warmup 2. Consider the polynomial

Let the coefficient of be. Find.

Warmup 3. Call a set of integers spacy if it contains no more

than one out of any three consecutive integers. How many

subsets of are spacy?

JUST A BIT MORE OF THE

BASICS!

You are given these two generating sequences:

Find and.

CONSIDER THIS PROBLEM…

Problem 1. Julia has a basket containing 4 apples and 5

bananas. Julia wants an even number of apples and at

least two bananas. In how many ways can Julia choose 6

pieces of fruit? (The apples are distinguishable, as are

the bananas.)

BUT WHAT IF…

Problem 1B. Julia

has a basket

containing 4 apples

and 5 bananas.

Julia wants an even

number of apples

and at least two

bananas. In how

many ways can

Julia choose pieces

of fruit? (The apples

are distinguishable,

as are the

 Let be the number of ways Julia can choose

apples.

 Similarly, let be the number of ways Julia can

choose bananas.

 Let be the ways Julia can choose pieces of fruit.

If Julia chooses apples, then he chooses bananas.

CONTINUED…

Problem 1B. Julia

has a basket

containing 4 apples

and 5 bananas.

Julia wants an even

number of apples

and at least two

bananas. In how

many ways can

Julia choose pieces

of fruit? (The apples

are distinguishable,

as are the

 Let be the ways Julia can choose pieces of fruit.

If Julia chooses apples, then he chooses bananas.

 He can choose apples and bananas in a total of

ways.

 Hence, since is the ways to choose fruits,

 appears to be the th term of the product!

THE ANSWER IS HERE!

Problem 1B. Julia

has a basket

containing 4 apples

and 5 bananas.

Julia wants an even

number of apples

and at least two

bananas. In how

many ways can

Julia choose pieces

of fruit? (The apples

are distinguishable,

as are the

 Let be the number of ways Julia can choose

apples and let be the number of ways Julia can

choose bananas. What is the generating function

for and?

 Problem 1C. In how many ways can Julia choose

pieces of fruit if? What are the possible values

of that yields a nonzero answer?

THE PRODUCT FORMULA

Turns out the “product of the generating function” argument

we used is an actual mathematical theorem.

Theorem 1. (The Product Formula) We are given two

disjoint sets and. We are also given that the number of ways

to choose elements from is , and the number of ways to

choose elements from is. Let be the number of ways to

choose elements (as a subset) from. Let , and be the

generating functions of the sequences , , and , respectively.

Then

THE BINOMIAL THEOREM IS

HERE!

Use the Binomial Theorem to help you with the

problem below.

Problem 5. Alvin is ordering 3 hot dogs at the snack

bar. He can choose among five toppings for each hot

dog: ketchup, mustard, relish, onions, and sauerkraut.

In how many different ways can he choose 6 toppings

total for his three hot dogs?

A QUICK SOLUTION

Problem 5. Alvin

is ordering 3 hot

dogs at the snack

bar. He can choose

among five

toppings for each

hot dog: ketchup,

mustard, relish,

onions, and

sauerkraut. In how

many different

ways can he

choose 6 toppings

 Since there is ways to get toppings for one hot

dog, the generating function for the scenario with

one hot dog would be

 However, since there are three hot dogs, the

generating function for the situation with three

hot dogs would be

 Hence, the number of ways Alvin can choose 6

toppings would simply be.

AN ALGEBRA INTERLUDE

 Problem 8. Find an expression for the coefficients of in the

infinite polynomial. (Hint: try things out!)

 Solution. Consider the cases when written in binomial

coefficients:

 We can intuitively conjecture that

 Looks familiar? Use distributions to explain why this statement is

valid.

YOUR TURN!

Problem 9. Ms. Connor is giving out identical pieces of candy to

the 5 officers. Joseph and Kelsey, being sophomores, want at most

1 piece. Alvin will take any number of pieces. However, Daniel and

Julia each demands an odd number of pieces. How many ways can

Ms. Connor distribute the candy?

Problem 10. Using distributions, explain why

Problem 11. Julia randomly selects two distinct numbers from the

set and Alvin randomly selects a number from the set. What is the

probability that Alvin’s number is larger the sum of the two

numbers chosen by Julia?