Arrangements

1
Arrangements Selections with Repetition
2
Arrangements with Unlimited Repetition

• Enumerating r-permutations from a set of n
  objects with repetition, denoted, U(n, r) nr.
• What does this number remind you of?
• Example
• There are 25 true/false questions on an
  examination.
• How many different ways can a student fill in
  answers, if she can also leave the answer blank?

3
Arrangements with Limited Repetition

• Generalizing the MISSISSIPPI example
• Theorem 1 If there are
• r1 objects of type 1,
• r2 of type 2, ,
• rm of type m,
• where r1 r2 . . . rm n,
• then the of arrangements of these n objects,
  denoted P(n r1 , r2 , , rm ) , is

4

• We have n positions to distribute the objects.
• Select r1 of them to position the objects of type
  1.
• n - r1 positions remain with rm-1 types of
  objects. Repeat. (Use an induction argument)

5
Proof

• Use the product rule
• Phase 1 pick the positions for the type 1
  objects
• There are nCr1 ways to do that.
• Phase 2 pick the positions for the type 2
  objects
• There are (n - r1)Cr2 ways to do that. And so
  on
• Phase m pick the positions for the type m
  objects
• There are (n - r1 - r2 - - rm-1)Crm ways to do
  that.

6
Example

• How many ways can 23 different books be given to
  5 students so that 2 students get 4 books each,
  and the other 3 get 5 books each?
• Use the product rule
• 1. Pick the 2 students of 5 who receive 4 books.
• 2. Distribute the books to the students.
• I.e., pick the 4 books for student1, pick the 4
  books for student2, pick the 5 books for
  student3,
• The answer thus is 5C2 ? P(234,4,5,5,5).

7
Example

• How many 8-digit sequences are there involving
  exactly 6 different digits?
• Use the product rule
• 1. Pick the 6 digits to be used.
• 2. For any given 6 digits, count the sequences

8

• For a given 6 digits, count the sequences
• Use the sum rule
• Partition the answers according to the 2 possible
  distribution of the 6 digits.
• 1 digit is used 3 times the other 5 are used
  once
• 2 digits are used twice the other 4 are used
  once
• Count the solutions for each distribution.
• Pick the digit to be used 3 times arrange the
  digits.
• Pick the digits to be used twice arrange the
  digits.

9
Selections with Unlimited Repetition

• Key to this kind of problem Knowing how to count
  the number of n-bit binary strings with exactly r
  1s nCr.
• We want to count the number of distinct
  selections of 12 items from identical
• Palm Pilots
• Nokia cell phones
• IBM Think Pads
• Uzi machine guns.

10

• For example, 1 possible such selection is
• 1 Palm Pilot
• 0 Nokia cell phones
• 10 IBM Think Pads
• 1 Uzi machine gun
• As a string, we could represent this selection
  as
• P//IIIIIIIIII/U

11

• How would we represent as a string the selection
• 2 Palm Pilot
• 3 Nokia cell phones
• 6 IBM Think Pads
• 1 Uzi machine gun
• If we agree on an order of item types, we could
  represent this selection as 00/000/000000/0

12

• Each string with 12 0s and 3 /s corresponds to 1
  selection.
• Each such selection corresponds to 1 such string.
• Thus, the problem is equivalent to asking How
  many (12 4 - 1)-bit binary strings are there
  with exactly 3 1s?
• There are (12 4 - 1)C3 (12 4 - 1)C12 such
  strings.

13
Selection with Unlimited Repetition Equivalent
Formulations

• The number of
• r-combinations of n distinct objects with
  unlimited repetition.
• nonnegative integer solutions to
  x1 x2 . . . xr n.
• ways to distribute n balls into r numbered boxes.
• (n r - 1)-bit binary strings with exactly r-1
  1s.
• (n r - 1)C(r -1) (n r - 1)Cn

14
Example

• How many ways are there to pick 10 balls from
  unlimited piles of identical red, blue, and black
  balls, and 1 green, 1 orange, and 1 yellow ball?

15
Example

• How many ways are there to pick 10 balls from
  unlimited piles of identical red, blue, and black
  balls, and 1 green, 1 orange, and 1 yellow ball?
• Partition the solution set into those that use 1,
  2, or 3 of the limited edition (LE) balls.
• For a given part, use the product rule
• count the ways to pick the LE balls
• count the ways to pick the rest of the balls.

16
Example

• How many ways are there to pick 20 recreational
  drug items from
• beers,
• joints,
• jolts,
• original bottles of coke (which contained
  cocaine)
• If you must have at least 2 beers, 3 joints, 1
  jolt, 3 cokes?

17

• This is equivalent to asking how many different
  selections are there of 20 - 2 - 3 - 1 - 3 11
  items (11 4 - 1)C(4 - 1).