Download Worksheet 4

MAT 142 College Mathematics
Module SC
Worksheet 4
Terri Miller
revised February 1, 2011
1. To fulfill certain requirements for a degree, a student must take one course each from
the following groups: health, civics, critical thinking, and elective. If there are four
health, three civics, six critical thinking, and ten elective courses, how many different
options for fulfilling the requirements does a student have?
Solution: 4 × 3 × 6 × 10 = 720
2. Calculate each of the following
(a) 6!
Solution: 720
(b) 8! ∗ 5!
Solution: 40320 × 120=4,838,400
9!
by expanding and simplifying
(c)
5!4!
Solution:
9!
9 · 8 · 7 · 6 · 5!
9·8·7·6
9·2·7·6
9·7·6
9·7·2
=
=
=
=
=
= 126.
5!4!
5! · 4 · 3 · 2 · 1
4·3·2·1
3·2·1
3·1
1
n!
, where n = 12 and r = 7
(d)
(n − r)!
Solution:
1
12!
12!
=
= 3, 991, 680
(12 − 7)!
5!
c 2011 ASU School of Mathematical & Statistical Sciences and Terri L. Miller
3. A nickel, a dime and a quarter are tossed,you observe the outcome for each coin (heads
or tails). Construct a tree diagram to list all possible outcomes. Use the Fundamental
Counting Principle to determine how many different outcomes are possible.
Solution:
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HT T
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YeeYYYYY
YYYYYY
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T HH
7H
ooo
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T YYYYYYYYYY
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TTH
TTT
number of outcomes: 2 × 2 × 2 = 8
4. How many different Zip Codes are possible using:
(a) the old style (five digits)
Solution: 10 × 10 × 10 × 10 × 10 = 105
(b) the new style (nine digits)
Solution: 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 109
(c) the old style if it cannot start with a zero
Solution: 9 × 10 × 10 × 10 × 10 = 9 × 104 = 90, 000
2
c 2011 ASU School of Mathematical & Statistical Sciences and Terri L. Miller
5. Each student and State University has a student ID number consisting of four digits
(the first digit is nonzero and digits may be repeated) followed by three of the letters
A, B, C, D, and E (letters may not be repeated). How many different student IDs are
possible?
Solution: 9 × 10 × 10 × 10 × 5 P3 = 150
6. List all the permutations of {a, b, c} when the elements are taken two at a time.
Solution: ab, ac, ba, bc, ca, cb
7. List all the combinations of {a, b, c} when the elements are taken two at a time.
Solution: ab, cb, ca (Note: each of the pairs can be written in either order)
8. There are 8 different books on a table. You are to line up 3 of them. How many
possible ways are there to do so?
Solution: Lining up implies that order matters, so 8 P3 = 336
9. There are 8 different books on a table. You get to select 3 of them. How many possible
choices are there?
Solution: Selecting a group of 3 implies that order does not matter, so 8 C3 = 56.
10. What is the formula to compute P (n, r)?
Solution: P (n, r) =
n!
(n − r)!
11. Evaluate P (7, 3) by using the formula and without the formula using your calculator.
Solution: using formula: P (7, 3) =
calculator: 7
n Pr
7!
7!
7 × 6 × 5 × 4!
=
=
= 7 × 6 × 5 = 210
(7 − 3)!
4!
4!
3 = 210
12. Find C(3, 2)
Solution: 3
3
c 2011 ASU School of Mathematical & Statistical Sciences and Terri L. Miller
13. A group of ten seniors, eight juniors, five sophomores, and five freshmen must select
a committee of four. How many committees are possible if the committee of 4 must
contain the following?
(a) one person from each class
Solution: 10 × 8 × 5 × 5 = 2000
(b) any mixture of the classes
Solution:
28 C4
= 20, 475
(c) exactly two seniors
Solution:
10 C2
× 18 C2 = 45 × 153 = 6885
14. How many five-card poker hands consisting of three kings and two queens are possible?
Solution: 4 C3 × 4 C2 = 4 × 6 = 24
15. A 7/39 lottery requires choosing seven of the numbers 1 through 39. How many
different lottery tickets can you choose? (Order is not important, and the numbers do
not repeat.)
Solution:
4
39 C7
= 15, 380, 937
c 2011 ASU School of Mathematical & Statistical Sciences and Terri L. Miller