Mathematics for Computer Science
MIT 6.042J/18.062J
More Counting
Albert R Meyer, November 16, 2009
lec 11M.1
Sum Rule
|A ∪ B| = |A| + |B|
A
B
for disjoint sets A, B
Albert R Meyer, November 16, 2009
lec 11M.2
Sum Rule
|A ∪ B| = ?
A
B
What if not disjoint?
Albert R Meyer, November 16, 2009
lec 11M.3
Inclusion-Exclusion
|A  B| =|A|+|B|-|A  B|
A
B
What if not disjoint?
Albert R Meyer, November 16, 2009
lec 11M.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, November 16, 2009
lec 11M.8
Incl-Excl Formula: Proof
A town has n clubs.
Each club Si has a secretary
Mi who knows if person x is
a club member:
Mi(x) = 1 if x in Si,
= 0 if x not in Si.
Albert R Meyer, November 16, 2009
lec 11M.10
Incl-Excl Formula: Proof
Fact: |A| = sum of MA(x) over all x.
Now Mi(x)Mj(x) is sec’y for Si∩Sj
so |Si∩Sj| =
sum of Mi(x)Mj(x) over all x,
|Si∩Sj∩Sk| =
x Mi(x)Mj(x)Mk(x)
etc.
Albert R Meyer, November 16, 2009
lec 11M.11
Incl-Excl Formula: Proof
Let D ::= S1∪S2∪ ∪Sn
sec’y MD(x)=0 iff Mi(x)=0
for all n clubs. So
1-MD(x) =
(1-M1(x))(1-M2(x)) (1-Mn(x))
Albert R Meyer, November 16, 2009
lec 11M.12
Incl-Excl Formula: Proof
1-MD(x) = (1-M1(x))(1-M2(x))
so...
MD(x) = i Mi(x)
-
i<jMi(x)Mj(x)
+
i<j<k
(1-Mn(x))
Mi(x)Mj(x)Mk(x)
+ (-1)n+1 M1(x)M2(x)
Albert R Meyer, November 16, 2009
Mn(x)
lec 11M.13
Incl-Excl Formula: Proof
1-MD(x) = (1-M1(x))(1-M2(x))
now sum both sides over x
MD(x) = i Mi(x)
-
i<jMi(x)Mj(x)
+
i<j<k
(1-Mn(x))
Mi(x)Mj(x)Mk(x)
+ (-1)n+1 M1(x)M2(x)
Albert R Meyer, November 16, 2009
Mn(x)
lec 11M.14
Incl-Excl Formula: Proof
1-MD(x) = (1-M1(x))(1-M2(x))
now sum both sides over x
|D| = i |Si|
-
i<jMi(x)Mj(x)
+
i<j<k
(1-Mn(x))
Mi(x)Mj(x)Mk(x)
+ (-1)n+1 M1(x)M2(x)
Albert R Meyer, November 16, 2009
Mn(x)
lec 11M.15
Incl-Excl Formula: Proof
1-MD(x) = (1-M1(x))(1-M2(x))
now sum both sides over x
|D| = i |Si|
-
i<j|Si∩Sj|
+
i<j<k
(1-Mn(x))
Mi(x)Mj(x)Mk(x)
+ (-1)n+1 M1(x)M2(x)
Albert R Meyer, November 16, 2009
Mn(x)
lec 11M.16
Incl-Excl Formula: Proof
1-MD(x) = (1-M1(x))(1-M2(x))
now sum both sides over x
|D| = i |Si|
-
i<j|Si∩Sj|
+
i<j<k
(1-Mn(x))
|Si∩Sj∩Sk|
+ (-1)n+1 M1(x)M2(x)
Albert R Meyer, November 16, 2009
Mn(x)
lec 11M.17
Incl-Excl Formula: Proof
1-MD(x) = (1-M1(x))(1-M2(x))
now sum both sides over x
|D| = i |Si|
-
i<j|Si∩Sj|
+
i<j<k
(1-Mn(x))
|Si∩Sj∩Sk)|
+ (-1)n+1 |S1∩S2 ∩Sn|
Albert R Meyer, November 16, 2009
lec 11M.18
Incl-Excl Formula: Proof
|D| =
+
|Si|
i<j|Si∩Sj|
i
i<j<k
|Si∩Sj∩Sk|
+ (-1)n+1 |S1∩S2 ∩Sn|
Albert R Meyer, November 16, 2009
lec 11M.19
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, November 16, 2009
lec 11M.20
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, November 16, 2009
lec 11M.21
Team Problems
Problems
1—3
(& maybe 4)
Albert R Meyer, November 16, 2009
lec 11M.22