Math 166 Week-in-Review - © S. Nite 9/29/2012
WIR #4
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Week in Review #4 (2.1-2.2)
The Multiplication Principle:
Suppose there are m ways of performing a task T1 and n ways of performing a task
T2. Then, there are mn ways of performing the task T1 followed by the task T2.
A permutation of a set of distinct objects is an arrangement of these objects in a
definite order.
For any natural number n, n-factorial is given by
n! = n(n – 1)(n – 2)⋅ . . . ⋅ 3 ⋅ 2 ⋅ 1
0! = 1
The number of permutations of n distinct objects taken n at a time:
P(n, n) = n!
The number of permutations of n distinct objects taken r at a time:
P(n, r) =
n!
(n − r )!
Permutations of n Objects, Not All Distinct:
Given a set of n objects in which n1 objects are alike and of one kind, n2 objects
are alike and of another kind,…, and, finally, nr objects are alike and of yet
another kind so that
n1 + n2 + ... + nr = n
then the number of permutations of these n objects taken n at a time is given by
n!
n1!n 2 !...n r !
A combination is a subset of r objects taken from a set of n objects, without any
regard to the order in which the objects are selected.
The number of combinations of n distinct objects taken r at a time is given by
C(n, r) =
n!
r!(n − r )!
(where r ≤ n)
Math 166 Week-in-Review - © S. Nite 9/29/2012
WIR #4
Page 2 of 5
1. An hamburger place has three flavors of ice cream (vanilla, chocolate, and
strawberry) and four toppings (butterscotch, fudge, caramel, and cherry). How
many different choices of ice cream with topping are there? What is the set of all
possible choices?
2. A group of four students working on a project must choose a leader and a
reporter. How many ways can this be done? What is the set of all possible
outcomes?
3. Eight teams are playing in a basketball tournament. For a competition, Jill must
choose a first place and a second place pick. How many possible sets of picks of
two teams can she make?
4. Eight teams are playing in a basketball tournament. For a competition, Jill must
choose a first place, a second place, and a third place pick. How many possible sets
of picks of three teams can she make?
Math 166 Week-in-Review - © S. Nite 9/29/2012
WIR #4
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5. How many codes with two letters followed by three numbers with no repetitions
of letters are possible?
6. A furniture store selects seven different recliners from a group of ten and has
each recliner shipped on successive weeks. How many ways can this be done?
7. A picture is to be taken by lining up 4 of the 10 percussionists, then 5 of the 7
trumpeters, and then 2 of the 6 trombonists. How many ways can this arrangements
be done?
8. How many different 5-card hands can be dealt from a standard deck of 52 cards?
9. How many different 5-card hands with two aces and three kings can be dealt
from a standard deck of 52 cards?
10. In how many ways can a committee be formed with a chair, reporter, and
treasurer, and 5 additional people, all chosen from a group of 12 people?
Math 166 Week-in-Review - © S. Nite 9/29/2012
WIR #4
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11. A committee of 18 people consists of 10 women and 8 men. In how many ways
can a subcommittee of four be formed if the subcommittee consists of any four
committee members?
12. A committee of 18 people consists of 10 women and 8 men. In how many ways
can a subcommittee of four be formed if the subcommittee consists of all women?
13. A committee of 18 people consists of 10 women and 8 men. In how many ways
can a subcommittee of four be formed if the subcommittee consists of at least four
men?
14. In Scotland, juries may consist of 15 people. If a jury is to be selected at
random from 25 people, and then 2 additional alternate jurors are selected, how
many ways can this be done?
15. A salesman has 12 customers in Philadelphia, 7 customers in Pittsburgh, and 4
customers in Millersville. In how many ways can she see 5 customers in
Philadelphia, 4 customers in Pittsburgh, and 2 customers in Millersville?
Math 166 Week-in-Review - © S. Nite 9/29/2012
WIR #4
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1, 1
1, 2
1, 3
1, 4
1, 5
1, 6
2, 1
2, 2
2, 3
2, 4
2, 5
2, 6
3, 1
3, 2
3, 3
3, 4
3, 5
3, 6
4, 1
4, 2
4, 3
4, 4
4, 5
4, 6
5, 1
5, 2
5, 3
5, 4
5, 5
5, 6
6, 1
6, 2
6, 3
6, 4
6, 5
6, 6