Page 171: Letters ABCDE, forming strings of length 3
34. With
repetitions: 5^3
Five choices for first letter, five choices for second letter, five choices for
third letter
35. Without
repetitions: 5*4*3
Five choices for first letter, four choices for second letter, three choices
for third letter
36. Beginning
with A, with repetitions: 5^2
1 choice (A) for first letter, five choices for second letter, five choices for
third letter
37. Beginning
with A, without repetitions: 4*3
1 choice (A) for first letter, four choices for second letter, three choices
for third letter
38. Without
A, with repetitions: 4^3
Selecting from {B, C, D, E}: Four choices for first letter, four choices for
second letter, four choices for third letter
39. Without
A, without repetitions: 4*3*2
Selecting from {B, C, D, E}: Four choices for first letter, three choices for
second letter, two choices for third letter
40. With
A, with repetitions: 5^3-4^3
All possibilities, minus the ones not containing A
Consider why this is wrong: Pick a position for A (3 ways); for the remaining
two positions, pick any of the 5 letters (5^2)
Hint: there is some double-counting – why?
41. With
A, without repetitions: 5*4*3 – 4*3*2 = 3*4*3
All possibilities, minus the ones not containing A
Alternate: Pick a position for A (3 ways); pick any of the remaining letters
for the first available position (4 ways); pick any of the remaining letters
for the final position (3 ways)
Page 172: Five computer science books, three math books, and two art books
54. All orderings: 10!
55. All
orderings, putting CS books on left, math books in center, and art books on
right: 5!*3!*2!
There are 5! ways to order the CS books
There are 3! ways to order the math books
There are 2! ways to order the art books
56. All
orderings, putting CS books on left: 5!*5!
There are 5! ways to order the CS books
There are 5! ways to order the remaining books
57. 3!*5!*3!*2!
a. Arrange the subjects (3!)
b. Arrange the cs books (5!)
c. Arrange the math books (3!)
d. Arrange the art books (2!)
58. 8!*C(9,2)*2!
a. Arrange the cs and math books (8!)
b. Pick which two of the 9 positions between (and outside) the previously arranged books to use for art books C(9,2)
c. Pick which art book goes in each position (2!)
59.
60. Functions
from X to Y, with |X| = n and |Y| = m: mn
Suppose that X = {x1, …, xn}. We need to pick a y value for x1 (there are m
ways to do it), a y value for x2 (there are m ways to do it), etc., repeated n
times.
Page 182: Number of strings formed by ordering the letters
ABCDE:
(Note that “ordering” implies “no repetitions”)
10. Containing
ACE: 3!
Permutations of { ACE, B, D}
11. Containing A, C, and E together in any order:
a. Order A, C, E: 3!
b. Order p(ACE) with B and D: 3!
c. Multiplication: 3!*3!
12. Containing DB and AE: 3!
13. Either AE or EA: 4!*2
14. A appears before D: C(5,2)*3! = 5!/(3!*2!)*3! = ½*5!
15. Neither AB nor CD:
a. Contains AB: 4!
b. Contains CD: 4!
c. Contains both AB and CD: 3!
d. Contains either AB or CD: 2*4! – 3!
e. Contains neither: 5! – (2*4! – 3!)
16. Containing neither AB nor BE:
a. Contains AB: 4!
b. Contains BE: 4!
c. Contains both: 3!
d. Contains either AB or BE: 2*4! – 3!
e. Doesn’t contain either 5! – (2*4! – 3!)
17. A before C and C before E: C(5,3)*2!
18. Either DB or BE (or both):
a. DB: 4!
b. BE: 4!
c. Both: 3!
d. Either: 2*4! – 3!
===================================================
Page 183: Club of six men and six women:
31. Committee of five: C(12,5)
32. Three men, four women: C(6,3)*C(6,4)
33. Four persons with at least one woman:
a. Four men: C(6,4)
b. Four persons: C(12,4)
c. Subtract: C(12,4) – C(6,4)
34. Four persons with at most one man:
a. Four women: C(6,4)
b. Three women and a man: C(6,3)*C(6,1)
c. Addition: C(6,4)+C(6,3)*C(6,1)
35. Four persons with at least one of each sex: C(12,4) – 2*C(6,4)
36. Four persons, not containing both Mabel and Ralph
a. Four persons, containing both: C(10,2)
b. Subtract: C(12,4)-C(10,2)
37. 4/10
GOP, 3/12 Dem., 2/4
38. C(8,3)
39. Three zeros in a row in an 8-bit string: 6
40.
Poker hands
41. Four aces: 48
42. Four of a kind: 13*48
43. All spades: C(13,5)
44. Cards of exactly two suits: C(4,2)*C(26,5)
45. Cards of all suits: Not restricted to any three suits
a. Cards of no suits: 0
b. Cards of 1 suit: C(4,1)*C(13,5)
c. Cards of two suits: C(4,2)*C(26,5)
d. Cards of three suits: C(4,3)*C(39,5)
e. None of the above: C(52,5) – C(4,3)*C(39,5) – C(4,2)*C(26,5) – C(4,1)*C(13,5)
46. A2345 in one suit: 4
47. Consecutive and of the same suit: 9*4 (A low)
48. 9*4^4
49. 13*C(4,2)*12*C(4,2)*11 ?*C(3,2)
Bridge hands
50.
Page 209: Number of strings by re-ordering letters
Comics are Action, Superman, Captain Marvel, Archie, X-man, Nancy
Categories
Bicycles
Books
Red, blue, and green balls
x1+x2+x3=15