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Math 127 Assignment #7 solutions

1. An exam has 10 true/false questions (T or F) and 6 multiple choice questions with 5

possible answers each.a)  If you have to answer all questions, how many possible different answer sheets

could you hand in? 210

56

= 1024 x 15625 = 16,000,000b)  If you only have to answer 6 T or F questions and 4 multiple choice questions, how

many possible choices do you have of selecting the questions to answer?

C(10,6)C(6,4) = 210x15 = 3,150

c)  If you only have to answer 6 T or F questions and 4 multiple choice questions, how

many possible different answer sheets could you hand in? C(10,6)26C(6,4)54 =

126,000,000

d)  In light of c), if you didn’t study and randomly guessed the answer to your

particular choice of questions, what is the probability that you got them all correct?

1/126,000,000 = .0000794…

e)  Suppose that you don’t read instructions and having got the exam paper you just

answer some of the questions; how many ways could you select the questions toanswer? How many subsets does a set of size 16 have? 216 = 65536.

2.  You're going to buy 40 cans of pop for your grade 6 class. There are Coke, Pepsi, 7-Up and Root Beer available. How many possible orders can you place if 

a) You place an order, such as 10 Cokes, 3 Pepsis, etc. with no restrictions? How

many ‘words’ can we make from 40 x’s and 3 |’s? C(43,3) = 12341.

b) You must have at least 5 Cokes, 2 Pepsis, 4 7-Up and 5 Root Beers in your order in

part a)? Order these 16 first; then how many words can we make from 24 x’s and

3 |’s? C(27,3) = 2925.

c) How many integer solutions are there to the equation c + p + s + r = 40 if you knowc 5, p 2, s 4 and r 5. This is b) in disguise: 2925.

3.  Before finals you have to complete 4 different Math assignments and 5 differentEnglish assignments. In how many ways can you do this if 

a)  there are no restrictions? 9! = 362,880

b)  you do all the Math assignments in a row? Think of the Maths as one object

temporarily and there are 6! Ways to arrange things; now account for the 4!

ways to arrange the Math assignments; all told 6!4! = 720x24 = 120,960 ways.

c) 

you never do two assignments of the same type in a row? Order the Maths in 4!ways; this determines 5 spots to place the English in 5! ways – all told 4!5! =

2,880 ways.

d)  somewhere along the line you do at least two assignments of the same type in a

row? This is a) – c) = 362,880 –2880 = 360,000 ways.

4. The following 'map' shows your home, H, and the university, U. Assuming that you

go directly from H to U, that is, you walk along the blocks to the right or up only,

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how many different ways can you walk to the university? Arrange 12 ’s and

4 ’s, C(16,4) = 1820

U

H

5. How many committees of size 5 can you form from 3 Liberals, 4 PCs and 5 NDPs if a)  there are no restrictions C(12,5) = 792

b)  there must be at least 2 PCs

If exactly 2 PC’s: C(4,2)C(8,3) = 6x56 = 336;

if exactly 3 PC’s: C(4,3)C(8,2) = 4x28 = 112;

if exactly 4 PC’s: C(4,4)C(8,1) = 1x8 = 8;

in total: 456c)  there must be at least one member of each party?

The number of (L,P,N)’s must be (3,1,1) or (1,3,1) or (1,1,3) or (2,2,1) or (2,1,2)

or(1,2,2).

In total we get C(3,3)C(4,1)C(5,1) + C(3,1)C(4,3)C(5,1) + C(3,1)C(4,1)C(5,3) +

C(3,2)C(4,2)C(5,1) + C(3,2)C(4,1)C(5,2) + C(3,1)C(4,2)C(5,2) = 1x4x5 + 3x4x5 +

3x4x10 + 3x6x5 + 3x4x10 + 3x6x10 = 20 + 60 + 120 + 90 + 120 + 180 = 590.