16.1 Fundamental Counting Principle

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16.1 Fundamental Counting Principle

• OBJ ? To find the number of possible
  arrangements of objects by using the Fundamental
  Counting Principle

2
DEF ? Fundamental Counting Principle

• If one choice can be made in a ways and a second
  choice can be made in b ways, then the choices in
  order can be made in a x b different ways.

3
EX ? A truck driver must drive from Miami to
Orlando and then continue on to Lake City. There
are 4 different routes that he can take from
Miami to Orlando and 3 different routes from
Orlando to Lake City.

• A
• C 1
• Miami G Orlando 7 Lake City
• T 9

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Strategy for Problem Solving

• 1) Determine the number of decisions.
• 2) Draw a blank (____) for each.
• 3) Determine of choices for each.
• 4) Write the number in the blank.
• 5) Use the Fundamental Counting Principle

• 2 Choosing a letter and a number
• 2) _____ _____
• 4 letters, 3_numbers
• 4__ 3__
• 4 x 3
• 12_ possible routes
• A1,A7,A9C1,C7,C9
• G1,G7,G9T1,T7,T9

5
EX ? A park has nine gatesthree on the west
side, four on the north side, and two on the east
side.

• 2 Choosing an entrance and an exit gate
• 1) 3 x 2
• west east
• 6
• 2) 4 x 4
• north north
• 16
• 3) 9 x 9
• enter leave
• 81

• In how many different ways can you
• 1) enter the park from the west side and later
  leave from the east side?
• 2) enter from the north and later exit from the
  north?
• 3)enter the park and later leave the park?

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EX ? How many three-digit numbers can be formed
from the 6 digits 1, 2, 6, 7, 8, 9 if no digit
may be repeated in a number

• 3 Choosing a 100s, 10s, and1s digit
• 6 x 5 x 4
• 100s 10s 1s
• 120

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EX ? How many four-digit numbers can be formed
from the digits 1, 2, 4, 5, 7, 8, 9

• if no digit may be repeated in a number?
• 4 Choosing a 1000s,100s,10s,1s digit
• 7 x 6 x 5 x 4
• 1000s 100s 10s 1s
• 840

• If a digit may be repeated in a number?
• 4 Choosing a 1000s,100s,10s,1s digit
• 7 x 7 x 7 x 7
• 1000s 100s 10s 1s
• 2401

8
EX ? How many three-digit numbers can be formed
from the digits 2, 4, 6, 8, 9 if a digit may be
repeated in a number?

• 3 Choosing a 100s, 10s, and1s digit
• 5 x 5 x 5
• 100s 10s 1s
• 125

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EX ? A manufacturer makes sweaters in 6
different colors. Each sweater is available with
choices of 3 fabrics, 4 kinds of collars, and
with or without buttons.

• How many different sweaters does the manufacturer
  make?
• 4 , ,
  , _
• color fabric collors
  with/without
• 6 x 3 x 4
  x 2
• 144

10
EX ? Find the number of possible batting orders
for the nine starting players on a baseball team?

• 9 decisions
• 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x
  1_
• 362,880
• (Also 9! Called 9 Factorial)

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16.2 Conditional Permutations

• OBJ ? To find the number of permutations of
  objects when conditions are attached to the
  arrangement.

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DEF ? Permutation

• An arrangement of objects in a definite order

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EX ? How many permutations of all the letters
in the word MONEY end with either the letter E or
the letter y?

• Choose the 5th letter, either a E or Y
• x ___ x ___ x ___ x 2
• 1st 2nd 3rd 4th 5th
• 4 x ___ x ___ x ___ x 2
• 1st 2nd 3rd 4th 5th
• 4 x 3 x 2 x 1 x 2
• 1st 2nd 3rd 4th 5th
• 48

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EX ? How many permutations of all the
letters in PATRON begin with NO?

• Choose the 1st two letters as NO
• 1 x 1 x __ x __ x __ x __
• 1st 2nd 3rd 4th 5th 6th
• 1 x 1 x 4 x __ x __ x __
• 1st 2nd 3rd 4th 5th 6th
• 1 x 1 x 4 x 3 x 2 x 1
• 1st 2nd 3rd 4th 5th 6th
• 24

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EX ? How many permutations of all the letters
in PATRON begin with either N or O?

• Choose the 1st letter, either N or O
• 2 x x __ x __ x __ x __
• 1st 2nd 3rd 4th 5th 6th
• 2 x 5 x x __ x __ x __
• 1st 2nd 3rd 4th 5th 6th
• 2 x 5 x 4 x 3 x 2 x 1
• 1st 2nd 3rd 4th 5th 6th
• 240

16

• NOTE From the digits
• 7, 8, 9, you can form 10
• odd numbers containing
• one or more digits if no
• digit may be repeated in
• a number.
• Since the numbers are
• odd, there are two
• choices for the units
• digit, 7 or 9.
• In this case, the numbers
• may contain one, two, or
• three digits.

• 1digit 7 9
• 2digit 79 87 89 97
• 3digit 789 879 897 987
• There are 2 one-digit numbers,
• 4 two-digit numbers,
• and 4 3 digit numbers.
• Since 2 4 4 10,
• this suggests that an or
• decision like one or more
• digits, involves addition.

17
EX ? How many even numbers containing one or
more digits can be formed from 2, 3, 4, 5, 6 if
no digit may be repeated in a number?

• Note there are three choices for a units
  digit 2, 4, or 6.
  
• X
• X X
• X X X
• X X X X

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EX ? How many odd numbers containing one or
more digits can be formed from 1, 2, 3, 4 if no
digit can be repeated in a number?

• X
• X X
• X X X

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NOTE In some situations, the total number of
permutations is the product of two or more
numbers of permutations. For example, there are
12 permutations of A, B, X, Y, Z with A, B to the
left and X, Y, Z to the right.

• ABXYZ ABXZY ABYXZ ABYZX
  ABZXY ABZYX
• BAXYZ BAXZY BAYXZ BAYZX BAZXY
  BAZYX
• Notice that
• (1) A, B can be arranged in 2!, or 2 ways
• (2) X, Y, Z can be arranged in 3!, or 6 ways and
• (3) A, B, X, Y, Z can be arranged in 2! x 3!, or
  12 ways.
• An and decision involves multiplication.

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EX ? Four different algebra books and three
different geometry books are to be displayed on a
shelf with the algebra books together and to the
left of the geometry books. How many such
arrangements are possible?

• ___X___ X___X____X____X ____X___
• ALG I ALG 2 ALG 3 ALG 4 GEOM I GEOM 2 GEOM
  3
• 4 X___ X___X____X 3 X ____X___
• ALG I ALG 2 ALG 3 ALG 4 GEOM I GEOM 2 GEOM
  3
• 4 X 3 X 2 X 1 X 3 X 2 X 1
  __
• ALG I ALG 2 ALG 3 ALG 4 GEOM I GEOM 2 GEOM
  3
• 144

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EX ? How many permutations of 1, A, 2, B, 3, C,
4 have all the letters together and to the right
of the digits?

• ___X___ X___X____X____X ____X___
• N 1 N 2 N 3 N 4 L 1 L 2 L 3
• 4 X___ X___X____X 3 X ____X___
• N 1 N 2 N 3 N 4 L 1 L 2 L 3
• 4 X 3 X 2 X 1 X 3 X 2 X
  1 _
• N 1 N 2 N 3 N 4 L 1 L 2 L 3
• 144