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 + r 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(23;4,4,5,5,5).

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 ( )-bit binary strings are there with exactly 3 1s?”
There are ( )C3 = ( )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 + x 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 ball[s]
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 = 11 items: ( )C(4 - 1).