1.10
1)
Problem Solving
with Permutations and Combinations
12 football players stand in a circular huddle. How many different arrangements of
the players are there?
or
2)
# with 2 letters + # with 3 letters
242 + 243
14400
In how many ways can 11 players be seated on the team bench so that Lindsay and
Rachael are not seated next to each other?
# ways L&R not together
5)
= total # - # without 2
=7x7x7-6x6x6
= 73 - 63
= 127
The Greek alphabet contains 24 letters. How many different Greek-letter fraternity
names can Jess create using either 2 or 3 letters? (Repetitions are allowed)
=
=
=
4)
3 991 680
How many 3-digit numbers can Doug form using only the numbers 1 to 7 if the number
2 must be included?
# with 2
3)
=
=total # ways - # ways L&R together
=11! - 10!2!
=32659200
In how many ways can 4 men and 4 women be seated around a circular table if each
man must be flanked by two women? (Hint: arrange the men first and then the women)
3!4!
or
=
144
6) Find the number of ways you can choose at least 1 piece of fruit from a
basket containing 4 apples, 5 bananas, 2 cantaloupes and 3 pears.
use
since at least one
=
=
5x6x3x4-1
359
7) In how many ways can you select a chairman, treasurer and secretary
from a board of directors with 8 members?
P(8,3) = 336
8) If 1000 people enter a contest in which there is a first prize, second
prize and third prize, in how many ways can the prizes be given?
P(1000,3) = 997 002 000
9) Chris and Shannon are starting out on their evening run. Their route
always takes them 8 blocks east and 5 blocks north to Mike‛s apartment
building. Chris likes to vary the path each night. How many different
possible routes does Chris have?
= 1287 or apply Pascal‛s Method
10) Jen and Lindsay are in charge of assigning rooms to the players on
the team. In how many ways can they assign the 12 basketball
players to 4 triple rooms?
= 369 600
11)
Danielle joins Cameron on his trip to the giant auction sale late in the afternoon. There
are only 5 items left to be sold. How many different purchases could Cameron make?
25 -1 = 31 or
12)
= 31
If Isabelle, Glen, Leah and Patricia play doubles matches in tennis, how many matches are
necessary if every player has every other player as a partner?
3! different teams but 2 teams / game, so
=3
13)
or
=3
A group of 25 students is flying to Akron, Ohio for their grad trip. There are 25 seats
available on the plane, 6 of which are first class. Alex and Heather won a draw and must
sit in first class. Rachel, Jenna and David are socially conscious and refuse to sit in first
class. With these restrictions in mind, in how many ways can the students be divided
between first class and economy?
6 for First Class, 19 econ - 2 FC seats taken so
only 4 left and 3 econ seats taken so only 16
left:
= 4845
14)
Richard wants to skateboard over to visit his friend Cathy who lives six blocks away.
Cathy‛s house is 2 blocks west and 4 blocks north of Richard‛s house. Each time Richard
goes over, he likes to take a different route. How many different routes are there for
Richard if he only travels west and north?
= 15 or apply Pascal‛s Method
15)
How many different sums of money can Justin form from one $2 bill, three $5 bills, two
$10 bills and one $20 bill?
16)
A 12-volume encyclopedia is to be placed on a shelf. In how many ways can Amanda
arrange them such that they are in an incorrect order?
# incorrect
17)
= total # - # correct
= 12! - 1
= 479 001 599
There are 12 questions on an examination, and Margie must answer 8 questions
including at least 4 of the first 5 questions. How many different combinations of
questions could she choose to answer?
= 210
18)
The 6 members of the yearbook staff sit around the circular table in their office.
How many different seating arrangements are there of this group of people?
or
19)
5! = 120
There are 8 runways at the regional airport. Pilots Jen, Alex and Jesse are each
bringing their planes in for a landing at approximately the same time. In how many
ways can air-traffic control sergeant Matt assign the planes to different runways?
P(8,3) = 336
20) A committee of 3 teachers is to select the winner from among 15 students nominated
for a special award. The teachers each make a list of their top 3 choices in order.
The lists have only 1 name in common, and the name has a different rank on each list.
In how many ways could the teachers have made the lists?
# choices for common name x # arrangements of common name x # arrangements remaining
= 15 x 3! x P(15,6)
= 324 324 000
P(15,1) x P(14,2) X P(12,2) x P(10,2) x 3! = 324 324 000