Mathematics for Computer Science
MIT 6.042J/18.062J
More Counting
Albert R Meyer, April 16, 2009
lec 10R.1
Sum Rule
|A  B| = |A| + |B|
A
B
for disjoint sets A, B
Albert R Meyer, April 16, 2009
lec 10R.2
Sum Rule
|A  B| = ?
A
B
What if not disjoint?
Albert R Meyer, April 16, 2009
lec 10R.3
Inclusion-Exclusion
|A  B| =|A|+|B|-|A  B|
A
B
What if not disjoint?
Albert R Meyer, April 16, 2009
lec 10R.4
Inclusion-Exclusion (3 Sets)
|A B C| =
|A|+|B|+|C|
– |A B| – |A C| – |B C|
+ |A B C|
A
B
C
Albert R Meyer, April 16, 2009
lec 10R.8
Incl-Excl Formula: Proof
A town has n clubs.
Each club Ai has a secretary
Si who knows if person x is
a club member:
Si(x) = 1 if x in Ai,
= 0 if x not in Ai.
Albert R Meyer, April 16, 2009
lec 10R.10
Incl-Excl Formula: Proof
Fact: |A| = sum of SA(x) over all x.
Now Si(x)Sj(x) is sec’y for AiAj
so |AiAj| =
sum of Si(x)Sj(x) over all x,
|AiAjAk| = x Si(x)Sj(x)Sk(x)
etc.
Albert R Meyer, April 16, 2009
lec 10R.11
Incl-Excl Formula: Proof
Let U ::= A1A2 An
sec’y SU(x)=0 iff Si(x)=0 for
all n clubs. So
1-SU(x) =
(1-S1(x))(1-S2(x)) (1-Sn(x))
Albert R Meyer, April 16, 2009
lec 10R.12
Incl-Excl Formula: Proof
1-SU(x) = (1-S1(x))(1-S2(x)) (1-Sn(x))
so...
SU(x) = i Si(x)
-
i<jSi(x)Sj(x)
+
i<j<k
Si(x)Sj(x)Sk(x)
+ (-1)n+1 S1(x)S2(x)
Albert R Meyer, April 16, 2009
Sn(x)
lec 10R.13
Incl-Excl Formula: Proof
1-SU(x) = (1-S1(x))(1-S2(x)) (1-Sn(x))
now sum both sides over x
SU(x) = i Si(x)
-
i<jSi(x)Sj(x)
+
i<j<k
Si(x)Sj(x)Sk(x)
+ (-1)n+1 S1(x)S2(x)
Albert R Meyer, April 16, 2009
Sn(x)
lec 10R.14
Incl-Excl Formula: Proof
1-SU(x) = (1-S1(x))(1-S2(x)) (1-Sn(x))
now sum both sides over x
|U| = i S
|A
i(x)
i|
-
i<jSi(x)Sj(x)
+
i<j<k
Si(x)Sj(x)Sk(x)
+ (-1)n+1 S1(x)S2(x)
Albert R Meyer, April 16, 2009
Sn(x)
lec 10R.15
Incl-Excl Formula: Proof
1-SU(x) = (1-S1(x))(1-S2(x)) (1-Sn(x))
now sum both sides over x
|U| = i S
|A
i(x)
i|
|A A |
-
i<jSi(x)S
j(x)
i
j
+
i<j<k
Si(x)Sj(x)Sk(x)
+ (-1)n+1 S1(x)S2(x)
Albert R Meyer, April 16, 2009
Sn(x)
lec 10R.16
Incl-Excl Formula: Proof
1-SU(x) = (1-S1(x))(1-S2(x)) (1-Sn(x))
now sum both sides over x
|U| = i S
|A
i(x)
i|
-
+
|A A |
Si(x)S
(x)S
(x)
A
A
|
i<j<k |A
j
k
i
j
k
i<jSi(x)S
j(x)
i
j
+ (-1)n+1 |A
S1(x)S
(x)
A
2
1
2
Albert R Meyer, April 16, 2009
S
A
n(x)
n|
lec 10R.17
Incl-Excl Formula
|U| =
+
S
|A
i(x)
i|
(x)
|A
A
i<jS
i(x)S
j
i
j|
i
i<j<k
Si(x)S
(x)S
(x)
|A
A
A
|
j
k
i
j
k
+ (-1)n+1 |A
S1(x)S
(x)
A
2
1
2
Albert R Meyer, April 16, 2009
S
A
n(x)
n|
lec 10R.18
bookkeeper rule
10!
# permutations of the word
2!2!3!
bookkeeper ?
• # perms bo1o2k1k2e1e2pe3r = 10!
• map perm o1be1o2k1rk2e2pe3 to
o1be
o2k1rk
rk2e2pe
pe3
obeokrkepe
1o
• 2 o’s, 2 k’s, 3 e’s:
map is 2!·2!·3!-to-1
Albert R Meyer, April 16, 2009
lec 10R.19
bookkeeper rule
# permutations of a word with
n1 a’s, n2 b’s, …, nk z’s is


n!
n

 ::=
n
,n
,
,n
n
!n
!
n
!
k 
 1 2
1 2
k
multinomial coefficient
Albert R Meyer, April 16, 2009
lec 10R.20
Team Problems
Problems
1—3
(& maybe 4)
Albert R Meyer, April 16, 2009
lec 10R.21