Counting Subsets
with Repetitions
ICS 6C
Sandy Irani
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}
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 .
Donut Selection
• Sample selection:
C C C G J J J J J J M M




3 chocolate
1 glazed
6 jelly
2 maple
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
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
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
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
Code Words to Donut Selections
• 001000101000000
• 100010000100000
• 110000000000001
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
12  4  1 15 
 
   
 4 1   3 
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  1
 

 m 1 
(This is still true if n > m, m < n, or m = n)
Balls into Bins
• How many ways to throw n identical balls into
m distinct bins?
Bin 1
Bin 2
Bin 3
Bin 4
CCGGGGJJJJJM
001000010000010
Counting Multisets
• How many ways to distribute 10 identical
prizes to a class with 200 students?
Solution to Sums of Variables
• How many solutions are there to the following
equation, where each variable xi is a non-negative
integer?
x1 + x2 + x3 + x4 = 12
001000010000010
x1 = 2
x2 = 4
x3 = 5
x4 = 1
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.
Solution to Sums of Variables
• How many solutions are there to the following
equation, where each variable xi is a non-negative
integer?
x1 + x2 + x3 + x4 = 12
x2 ≥ 1 and x4 ≥ 3 .
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
Chocolate Bar Distribution
• How many ways to distribute 15 identical
chocolate bars to 5 kids if each kid gets at
least one?
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)
Balls into Bins
• m distinguishable bins
• n balls
At most 1
per bin
distinguishable
balls
indistinguishable
balls
No limit on
number per bin
Same number
in each bin
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
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
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
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
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
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