Homework Answers

Section 4.1
2
 27*37 = 999
4
 72
6
 4*6 = 24
8
 26*25*24 = 15600
10
 2^8 = 256
12
 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 = 127
14
 2^(n-2)
16
 26^4 - 25^4 = 66351
22
 a.  10*9*8*7 = 5040,     b.  10*10*10*5,     c.  4*9 = 36
24
 2*(10*10*10*26*26*26) = 35152000
30
 a. 2^10,   b. 3^10,   c. 4^10,   d. 5^10
48
 TreeDiagram
Section 4.3
2
 7! = 5040
4
 a.  There are 60 of them, starting with "123, 124, ..., 213, 213, ..., ..."
b.  There are 10 of them.  "123, 124, 125, 134, 135, 145, 234, 235, 245, 345."
6
 a. 5,   b. 10,   c. 70,   d. 1,   e. 1,   f. 924
8
 5! = 120
10
 6! = 720
12
 C(99, 2) = 4851
14
 C(10, 1) + C(10, 3) + C(10, 5) + C(10, 7) + C(10,9) = 512
20
 C(40, 17).  (Don't bother with the exact value.)
26
 C(15, 4)*C(10, 2) + C(15, 5)*C(10,1) + C(15, 6)*C(10,0) = 96460
36
 C(13, 5) = 1287
44
 1  11  55  165  330  462  462  330  165  55  11  1
Section 4.6
18
 20! / (2!*4!*3!*2!*3!*2!*3!)
26
 11! / 4!*4!*2! = 34650
28
 Treat "AAA" as a single letter:  6! / 2! = 360
32
 C(14, 6) = 3003
34a
 40! / (10!*10!*10!*10!)
36
 16! / (4! * 3! * 5! * 4!) = 50450400
51
 10! / (3!*2!*5!) = 2520

Worksheet 1 Answers
1
 Done in class
2
 6! = 720
3
 17*22*2 = 748
4
 2^4 = 16 in each case,
5
 13 Bit Strings without "OO"
6
 1 letter:  6 ways
2 letters:  6*5 = 30 ways
1 or 2 letters:  6 + 30 = 36 ways
3, 4 or 5 letters:  6*5*4 + 6*5*4*3 + 6*5*4*3*2 = 1200 ways
We use the multiplication rule to find the number of ways for a given amount of letters.  In cases where there is more than one number of letters allowed, we add together the numbers of ways for each amount of letters
7
 There are 10*10*10 = 1000 combinations.  This takes 500 seconds.  In 9 minutes there are 540 seconds, which means he has 40 seconds to defuse the bomb, which will take him only 30 seconds.  So yes, he can make it.
8
 12 anagrams of REED
9
 b.  There are 24 ways
c.  Observe the following:  Each column can contain at most one rook, and since there are 4 rooks and 4 columns, we see that every column must contain exactly one rook.  Then we ask "where in each column shall we place our rooks?"  We have 4 choices for the first column, but then only 3 choices for the second column, since rooks may not share a row, 2 choices for the third column and 1 choice for the fourth column.  That's 4! = 24 ways.
10
 a.  5*5*5 = 125
b.  5*4*3 = 60
c.  10 ways.  Easy way is to build a tree diagram.  Clever way is to see that the answer is C(5, 3).
11
 a.  2^6 = 64
b.  3^4 = 81
c.  4^5 = 1024
d.  You need 2^x = 1024.   This is true when x = 10.  So the answer is 10.
12
 Same as Problem 8, for some reason
13
 5! / 2! = 60.
With the E's together, there would be 4! = 24, because we just think of "EE" as a single letter.
There are 60 - 24 = 36 ways which have the E's separated.
14
 The easiest way to do this is with a tree diagram.  Suppose the players are "1" and "2"
GameTree
There are 10 ways.  For four games, there are 14 ways.
15
 8^5 = 32768.  (We assume the dice are distinguishable.)
To get a "6", the dice must show 1, 1, 1, 1, 2.  There are 5 ways for this to happen.
To get a "7", the dice must show 1, 1, 1, 1, 3 (5 ways) or 1, 1, 1, 2, 2 (10 ways).  That's 15 ways total.
The chances of rolling a 6 or 7 is then (5 + 15) / 32768.
16
 Within divisions:  5!*5!*4!*5!*5!*4!
Within leagues:  14!*14!
Altogether:  28!
World Series matchups:  14*14 = 196.
17
 One.  The others were coming back.

Worksheet 2
2 Kentucky:  8! / 2! = 20160
Tennessee:  9! / 2!2!4! = 3780
3 C(9, 3)*C(12, 3)
4
 C(14, 3)
5
 LEVEE:  24
TEETH:  30
MONSOON:  420
MISSISSIPPI:  34650
6
 Upper left:  10
Bottom left:  210
Right:  C(20, 10)
7
 a. 6!        b. 20.    c. 120
8
 a.  26*26*26
b.  26*25*24
c.  C(26, 3) = 2600
9
 Select size 4, and they'll get 70 days-worth of delegations
10
 128, 21 and 21.
11
 6! / (2! 2! 2!) = 90
12
 8! * 11! in the firrst case, 19! in the second case
13
 a.  2775, 2700
b.  72, 75*72 = 5400, 5400/2 = 2700
14
 Let x be the number of men.  The number of handshakes is 21 * 5, which must equal x * 3.  So x = 35.