Counting Subsets with Repetitions. ICS 6C Sandy Irani

Transcription

1 Counting Subsets with Repetitions ICS 6C Sandy Irani

2 Multi-sets A Multi-set can have more than one copy of an item. Order doesn t matter The number of elements of each type does matter: {1, 2, 2, 2, 3, 4, 5, 7, 7} {1, 2, 2, 2, 3, 4, 7, 5, 7} {1, 2, 2, 3, 4, 5, 7, 7}

3 Counting Problem with Multi-sets You are purchasing a dozen donuts. The bakery has four varieties of donuts: Chocolate, Glazed, Jelly, Maple Donuts of the same variety are the same. There is an unlimited supply of each variety. (For this problem at least 12 of each variety). How many ways to select the donuts? Order doesn t matter. All that matters is how many of each variety you have in the box when you leave the bakery.

4 Donut Selection Sample selection: C C C G J J J J J J M M 3 chocolate 1 glazed 6 jelly 2 maple

5 Donut Selection Binary encoding that uniquely specifies a selection of donuts: Bijection: Selection of donuts Binary code word Will count the Number of valid binary code words

6 Encoding Donut Selection C C C G J J J J J J M M Select any ordering of the varieties: 1. Chocolate 2. Glazed 3. Jelly 4. Maple

7 Encoding Donut Selection G G G G J J J J J J M M Every code word has 12 0 s and 3 1 s: total of 15 bits

8 Encoding Donut Selection C C C C J J J J J J M M Every code word has 12 0 s and 3 1 s: total of 15 bits Number of donuts Number of varieties - 1

9 Code Words to Donut Selections

10 Donut Selection Bijection: Ways to select 12 donuts from 4 varieties Binary strings with 12 0 s and (4-1) 1 s # of ways to select 12 donuts from 4 varieties

11 Selecting from Varieties The number of ways to select n items from a set of m varieties Items from the same variety are identical There is at least n items from each variety n m m 1 1 (This is still true if n > m, m < n, or m = n)

12 Balls into Bins How many ways to throw n identical balls into m distinct bins? Bin 1 Bin 2 Bin 3 Bin 4 C C G G G G J J J J J M

13 Counting Multisets How many ways to distribute 10 identical prizes to a class with 200 students?

14 Solution to Sums of Variables How many solutions are there to the following equation, where each variable x i is a non-negative integer? x 1 + x 2 + x 3 + x 4 = x 1 = 2 x 2 = 4 x 3 = 5 x 4 = 1

15 Lower Bounds How many ways to select 12 donuts from 4 varieties (choc, glazed, jelly, maple) with the added constraint that there are at least 2 chocolate donuts? First pick the 2 chocolate donuts. Then select the remaining 10 donuts with no restrictions.

16 Solution to Sums of Variables How many solutions are there to the following equation, where each variable x i is a non-negative integer? x 1 + x 2 + x 3 + x 4 = 12 x 2 1 and x 4 3.

17 Balls into Bins How many ways to throw 12 identical balls into 4 distinct bins with at least two in each bin? Bin 1 Bin 2 Bin 3 Bin 4

18 Chocolate Bar Distribution How many ways to distribute 15 identical chocolate bars to 5 kids if each kid gets at least one?

19 Upper Bounds (by complement) How many ways to select 12 donuts from 4 varieties (choc, glazed, jelly, maple) with the added constraint that there are only 5 chocolate donuts available? The number of ways to select 12 donuts from 4 varieties with 5 chocolates The number of ways to select 12 donuts from 4 varieties with no constraints = - The number of ways to select 12 donuts from 4 varieties with NOT( 5 chocolates)

20 Balls into Bins m distinguishable bins n balls At most 1 per bin No limit on number per bin Same number in each bin distinguishable balls indistinguishable balls

21 Balls Into Bins How many ways are there to put 3 indistinguishable balls into 5 distinguishable bins with at most one per bin? Bin 1 Bin 2 Bin 3 Bin 4 Bin 5

22 Balls Into Bins How many ways are there to put 3 distinguishable balls into 5 distinguishable bins with at most one per bin? Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5

23 Balls Into Bins How many ways are there to put 3 distinguishable balls into 5 distinguishable bins with no limit on the number of balls per bin? Bin 1 Bin 2 Bin 3 Bin 4 Bin 5

24 Balls into Bins How many ways to throw n identical balls into m distinct bins? Bin 1 Bin 2 Bin 3 Bin 4 Bin 5

25 Balls into Bins How many ways to throw n distinct balls into m distinct bins same number of balls in each bin? Bin 1 Bin 2 Bin 3 Bin 4 Bin 5

26 Balls into Bins How many ways to throw n identical balls into m distinct bins same number of balls in each bin? Bin 1 Bin 2 Bin 3 Bin 4 Bin 5