C: ANSWERS TO SELECTED PROBLEMS
Chapter 3.1, Trees and Equally Likely Outcomes
1. S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.
3. S = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20,
24, 25, 30, 36}. Remember,
repeated
elements
are rostered only once in a set.
5. 
, S = {1H,
1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T}.
7. 
, S = {ABC,
ACB, BAC, BCA, CAB, CBA}.
9. There will be
three different outcomes on the tree diagram, S = {RB, BB, BR}.
11. 
13. 5,040.
15. 
.
17. 
19. 4! = 24.
21. 
.
23. Each of the 5
persons will shake hands with 4 other people. 
. However, this
includes a
second
handshake for each pair of people, so we divide this number in half. There are 10.
25. 
.
27. 
.
29. 
.
31. 
.
Chapter 3.2, Permutations
1. [a] 120, [b] 720, [c] 5, [d] 1.
3. P(20, 3) = 6,840.
5. P(7, 4) = 840.
7. There are 11
letters of which the M, A, and T
are each repeated twice, 
.
9. 
.
11. Fixing one seat
at the table converts the problem to 14 seats (1 vacant) in a row, or 13! ways.
13. 
.
15. Start with (8 -
1)! ways to place 8 keys on a ring (circular permutations) then multiply by 2
for
each
key (each key could face up or down).
Therefore, 
.
17. 
.
19. 
.
21. 
.
23. 
.
25. Start with the
total number of ways with no restrictions, 
, then subtract the number of
ways
with all evens, 0 (there is only 2 even elements); and subtract the number of
ways for
all
odds, 
. That is, 60 Ð
0 Ð 6 = 54.
27. 8 games are
required.
29. The row could
begin with a boy or a girl, 2 choices.
Then, P(5,5) ways to seat
the boys and
P(5,5) ways to seat the girls. 
.
31. 
.
Chapter 3.3, Combinations
1. [a] 1, [b] 210, [c] 1, [d] 6.
3. C(20, 3) = 1,140.
5. C(52, 5) = 2,598,960.
7. 
.
9. Read across the 6
th row of the triangle, [a] 1, [c] 10, [e] 5.
11. Either 1 B and 2
non-B or 2B and 1 non-B will satisfy, 
.
13. Need to pick 2
kinds, then pick which of the 2 will be 4 of kind, then pick 1 card of the
other
kind. 
.
15. P(31, 2) = 930.
17. 
19. 
.
21. 
.
23. 
25. C(30, 5) = 142,506.
27. [a] C(8, 4) = 70, [b] 
.
29. To pick the 8 in
favor (or 4 against) is 
.
Chapter 3.4, Chapter Review
Mastery Quiz
1. [a], 2. [b], 3. [a], 4.
[a], 5. [b], 6. [a], 7. [b], 8. [b], 9. [c], 10. [c]
Review
1. [a] 336, [b] 720, [c] 15, [d] 362,880, [e] 36.
3. [a] P(20, 3) = 6,840 [b] C(20, 3) = 1,140.
5. 
.
7. 
.
9. Cannot have 1 M
and 3 F because there are only 2 F, thus 2 M and 2 F or 3 M and 1 F,
.
11. First select
toppings, then crust, then drink, 
13. 
.
15. 
.
17. 
19. Construct a
chart, 5 ways.
21. 
23. Row starts with
men or with women, 
.
25. 
.
27. 
29. 
.
31. [a] Number of distinguishable arrangements
of 12 people is 12!, with pairing of 2 people per room being indistinguishable
is 
.
[b] Pick 3 of the 6 rooms for the men, arrange the 6 men
into 3 groups of 2, then arrange the 6 women into 3 groups of 2. 
33. 