Download Algebra III Lesson 45

Algebra III
Lesson 45
Conditional Permutation – Two-Variable
Analysis Using a Graphing Calculator
Conditional Permutations
Review:
How are permutations written?
Find:
8 P2
=
8!
(8 − 2)!
8!
=
6!
8 ⋅ 7 ⋅ 6!
=
6!
n
Pr
= 56
All that is meant by conditional is that maybe some outside
rules apply to what kinds of combinations are allowed.
There is no formula for doing these problems. You just have to
adjust and modify as needed by the problem.
Example 45.1
How many odd counting numbers can be formed from the digits 3,
4, and 5 if no repetition of digits is permitted?
No restriction on the length of the number.
1-digit
How many choices?
2
2-digit
2
2
3-digit
1
2
Total: 2
Which box should be filled first?
Right
How many choices?
How many choices for the other box?
Total: 4
Which box should be filled first?
Right
How many choices?
2
How many choices for the ten’s box?
How many choices for the hundred’s box?
Total: 4
Grand Total:
10
Example 45.2
Find the number of odd three-digit counting numbers that
are less than 600?
3 items
5
10
5
What are the conditions?
odd, <600
Start where?
doesn’t matter, repetition okay
Hundred’s box choices?
1,2,3,4,5
Ten’s box choices?
0,1,2,3,4,5,6,7,8,9
One’s box choices?
1,3,5,7,9
Total: 5·10·5 = 250
Example 45.3
Five math books and four English books are on a shelf. How many
permutations are possible if the math books must be kept together and
the English books must be kept together?
Math-English
5
4
3
2
1
4
3
2
1
How many choices for the first spot?
Total: 5!4!
And…
=2880
English-Math
4
3
2
1
5
4
3
2
1
How many choices for the first spot?
Total: 4!5!
Grand Total: 5760
And…
=2880
Example 45.4
How many four-digit odd counting numbers can be formed if no
repetition of digits is permitted?
4 items
8
8
7
5
Conditions?
odd, no repetition
Start where?
one’s box, nail down that first
So…
Total:
8·8·7·5
=2240
Example 45.5
An elf, a gnome, a fairy, a pixie, and a leprechaun were to sit
in a line. How many different ways can they sit if the elf and
the gnome insist on sitting next to each other?
5 items
E
G
Conditions?
3
2
1
elf and gnome next to each other.
Place the elf somewhere (first box for ease)
Gnome next.
And…
Total:
6
what about gnome first? Total:
6
Grand Total: 12
Two-variable analysis using a graphing calculator
and any of these problems in
assignments from now on.
Practice
a) How many three-digit counting numbers are there that
are less than 300 such that all the digits are even?
3 items
1
5
Conditions?
5
all even, <300
Start where?
So, 100’s, 10’s 1’s
Total: 25
anywhere
b) Write the equation of the following sinusoid:
10
270°
0°
90° 180°
-10
y=-10sinθ
360°
3
log10 10,000 = x
c) Solve for x:
4
Rewrite using power rule
log10 (10,000 ) = x
3
4
Rewrite without log
(10,000)
3
4
= 10 x
Regroup left to a power of 10
(10 )
3
4 4
= 10 x
Clean up and solve
103 = 10x
x=3
d) Find the domain of the function
f ( x) = 1− 4xx
From top
From bottom
x≥0
1 - |4x| ≠ 0
|4x| ≠ 1
So,
4x ≠ 1
Therefore,
4x ≠ -¼, ¼
⎧
⎨ x ∈ ℜ x ≥ 0, x ≠
⎩
1⎫
⎬
4⎭
4x ≠ -1