Arrangements & Selections with Repetition

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?

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

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)

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.

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).

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

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.

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.

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

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

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.

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

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?

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.

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?

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).