MATH 106 Combinatorics

1
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.
2
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
3
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!
4
(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!
5
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.
6
(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!
7
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!
8
(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!
9
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!
10
(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!
11
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!
12
(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!
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
14
(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
15
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
16
(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
17
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
18
(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
19
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
20
(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
21
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
22
(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
23
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
24
(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