Subsets, Strings, Equations Handout

NAME______________________________________________

____

A vending machine advertises Reeses Cups,

Hershey Bars, Snickers Bars, Milky Way Bars, and

Granola Bars, and each choice in the vending

machine costs 0.50.

1.

Use this class period to work on 1 on this

handout, which must be submitted for homework

either at the end of this class or in the class

indicated on the course schedule. As you work on

each problem, check to see if your final answer

is correct. We will go over in class any

problems there are questions about. There will

also be some time this period for questions.

Quiz 4 COMING UP! Be sure to do the review

problems for this, quiz posted on the internet.

The link can be found in the course schedule.

CHECK YOUR ANSWERS 1. (a) (b) (c) (d) (e)

(f) (g) (h) (i) (j) (k) (l) (m) (n)

(o) (p) (q) (r) (s) (t) (u) (v)

1

126

4845

969

56

1365

15

10

560

3025

2380

121

15

3844

111

425

2050

5

164

480

1610

179,180

Subsets, Strings, Equations Handout

NAME______________________________________________

____

A vending machine advertises Reeses Cups,

Hershey Bars, Snickers Bars, Milky Way Bars, and

Granola Bars, and each choice in the vending

machine costs 0.50.

1. (a)

How many different ways can Jane spend 2.50 at

the vending machine?

r h s m g 5 non-negative integers

9! 126 4! 5!

(b)

How many different ways can Sam spend 8.00 at

the vending machine?

r h s m g 16 non-negative integers

20! 4845 4! 16!

1.-continued (c)

How many different ways can Jane spend 2.50 at

the vending machine, if she wants to guarantee to

purchase at least one of each choice?

r h s m g 5 1 ? r 1 ? h 1 ? s

1 ? m 1 ? g OR positive integers

After giving 1 unit to each variable, we restate

the problem

r h s m g 0 . non-negative integers

There is only 1 (one) possible solution.

(d)

How many different ways can Sam spend 8.00 at

the vending machine, if he wants to guarantee to

purchase at least one of each choice?

r h s m g 16 1 ? r 1 ? h 1 ? s

1 ? m 1 ? g OR positive integers

After giving 1 unit to each variable, we restate

the problem

r h s m g 11 . non-negative integers

15! 1365 4! 11!

1.-continued (e)

How many different ways can Jane spend 2.50 at

the vending machine, if she does not want to

purchase any Granola Bars?

r h s m 5 non-negative integers

8! 56 3! 5!

(f)

How many different ways can Sam spend 8.00 at

the vending machine, if he does not want to

purchase any Granola Bars?

r h s m 16 non-negative integers

19! 969 3! 16!

1.-continued (g)

How many different ways can Jane spend 2.50 at

the vending machine, if she wants to purchase

exactly three Granola Bars?

r h s m g 5 g 3

After giving 3 units to g, we restate the

problem

r h s m 2 non-negative integers

5! 10 3! 2!

(h)

How many different ways can Sam spend 8.00 at

the vending machine, if he wants to purchase

exactly three Granola Bars?

r h s m g 16 g 3

After giving 3 units to g, we restate the

problem

r h s m 13 non-negative integers

16! 560 3! 13!

1.-continued (i)

How many different ways can Jane spend 2.50 at

the vending machine, if she wants to purchase at

least three Granola Bars?

r h s m g 5 g ? 3

After giving 3 units to g, we restate the

problem

r h s m g 2 non-negative integers

6! 15 4! 2!

(j)

How many different ways can Sam spend 8.00 at

the vending machine, if he wants to purchase at

least three Granola Bars?

r h s m g 16 g ? 3

After giving 3 units to g, we restate the

problem

r h s m g 13 non-negative integers

17! 2380 4! 13!

1.-continued (k)

How many different ways can Jane spend 2.50 at

the vending machine, if she wants to purchase at

most three Granola Bars?

r h s m g 5 g ? 3

We shall use the GOOD ALL BAD principle.

number of solutions in non-negative integers

9! 126 4! 5!

number of solutions in non-negative integers with

g ? 4

5! 5 4! 1!

The desired number of solutions is

126 5 121

(l)

How many different ways can Sam spend 8.00 at

the vending machine, if he wants to purchase at

most three Granola Bars?

r h s m g 16 g ? 3

We shall use the GOOD ALL BAD principle.

number of solutions in non-negative integers

20! 4845 4! 16!

number of solutions in non-negative integers with

g ? 4

16! 1820 4! 12!

The desired number of solutions is

4845 1820 3025

1.-continued (m)

How many different ways can Jane spend 2.50 at

the vending machine, if there are only two Milky

Way Bars left in the vending machine?

r h s m g 5 m ? 2

We shall use the GOOD ALL BAD principle.

number of solutions in non-negative integers

9! 126 4! 5!

number of solutions in non-negative integers with

m ? 3

6! 15 4! 2!

The desired number of solutions is

126 15 111

(n)

How many different ways can Sam spend 8.00 at

the vending machine, if there are only five Milky

Way Bars left in the vending machine?

r h s m g 16 m ? 5

We shall use the GOOD ALL BAD principle.

number of solutions in non-negative integers

20! 4845 4! 16!

number of solutions in non-negative integers with

m ? 6

14! 1001 4! 10!

The desired number of solutions is

4845 1001 3844

1.-continued (o)

How many different ways can Jane spend 2.50 at

the vending machine, if she wants to purchase at

least three Granola Bars and there are only two

Milky Way Bars left in the vending machine?

r h s m g 5 g ? 3 m ? 2

We shall use the GOOD ALL BAD principle.

6! 15 4! 2!

number of solutions in non-negative integers with

g ? 3

number of solutions in non-negative integers with

g ? 3 m ? 3

0

The desired number of solutions is

15 0 15

(p)

How many different ways can Sam spend 8.00 at

the vending machine, if he wants to purchase at

least three Granola Bars and there are only five

Milky Way Bars left in the vending machine?

r h s m g 16 g ? 3 m ? 5

We shall use the GOOD ALL BAD principle.

17! 2380 4! 13!

number of solutions in non-negative integers with

g ? 3

number of solutions in non-negative integers with

g ? 3 m ? 6

11! 330 4! 7!

The desired number of solutions is

2380 330 2050

1.-continued (q)

How many different ways can Jane spend 2.50 at

the vending machine, if she wants to purchase at

most three Granola Bars and at least four Reeses

Cups?

r h s m g 5 r ? 4 g ? 3

We shall use the GOOD ALL BAD principle.

5! 5 4! 1!

number of solutions in non-negative integers with

r ? 4

number of solutions in non-negative integers with

r ? 4 g ? 4

0

The desired number of solutions is

5 0 5

(r)

How many different ways can Sam spend 8.00 at

the vending machine, if he wants to purchase at

most three Granola Bars and at least eight

Reeses Cups?

r h s m g 16 r ? 8 g ? 3

We shall use the GOOD ALL BAD principle.

12! 495 4! 8!

number of solutions in non-negative integers with

r ? 8

number of solutions in non-negative integers with

r ? 8 g ? 4

8! 70 4! 4!

The desired number of solutions is

495 70 425

1.-continued (s)

How many different ways can Sam spend 8.00 at

the vending machine, if he wants to purchase at

least four, but no more than nine, Reeses Cups?

r h s m g 16 4 ? r ? 9

We shall use the GOOD ALL BAD principle.

16! 1820 4! 12!

number of solutions in non-negative integers with

r ? 4

number of solutions in non-negative integers with

r ? 4 r ? 10

10! 210 4! 6!

The desired number of solutions is

1820 210 1610

(t)

How many different ways can Sam spend 8.00 at

the vending machine, if he wants to purchase at

least one of each choice and at least four, but

no more than nine, Reeses Cups?

r h s m g 16 positive integers 4 ? r ?

9

We shall use the GOOD ALL BAD principle.

12! 495 4! 8!

number of solutions in positive integers with r ?

4

number of solutions in positive integers with r ?

4 r ? 10

6! 15 4! 2!

The desired number of solutions is

495 15 480

1.-continued (u)

Right next to the vending machine described

previously is a second vending machine

advertising a bag of potato chips, a bag of

pretzels, a bag of cheese doodles, and a bag of

cookies, each bag costing 0.50. How many

different ways can Jane spend 2.50 at both

vending machines, if she wants to purchase at

least two, but no more than four, bags of potato

chips?

r h s m g c p d k 5 2 ? c ? 4

We shall use the GOOD ALL BAD principle.

11! 165 8! 3!

number of solutions in non-negative integers with

c ? 2

number of solutions in non-negative integers with

c ? 2 c ? 5

8! 1 8! 0!

The desired number of solutions is

165 1 164

(v)

Right next to the vending machine described

previously is a second vending machine

advertising a bag of potato chips, a bag of

pretzels, a bag of cheese doodles, and a bag of

cookies, each bag costing 0.50. How many

different ways can Sam spend 8.00 at both

vending machines, if he wants to purchase at

least three, but no more than six, bags of potato

chips?

r h s m g c p d k 16 3 ? c ? 6

We shall use the GOOD ALL BAD principle.

21! 203,490 8! 13!

number of solutions in non-negative integers with

c ? 3

number of solutions in non-negative integers with

c ? 3 c ? 7

17! 24,310 8! 9!

The desired number of solutions is

203,490 24,310 179,180