Mathematics for Computer Science
MIT 6.042J/18.062J
Polynomials Express Choices & Outcomes
(
Binomial Theorem,
Combinatorial Proof
+
+
)(
+
+
+
+
)=
+
+
Image by MIT OpenCourseWare.
Products of Sums = Sums of Products
Albert R Meyer,
April 21, 2010
lec 11W.1
expression for ck?
Albert R Meyer,
April 21, 2010
lec 11W.2
expression for ck?
n times
multiplying gives 2n product terms:
111 + X11XX1 + 1XX1X1 ++ XXX
a term corresponds to selecting 1 or X
from each of the n factors
Albert R Meyer,
April 21, 2010
lec 11W.3
expression for ck?
Albert R Meyer,
April 21, 2010
lec 11W.5
The Binomial Formula
binomial
expression
n times
the Xk coeff, ck, is # terms
with exactly k X’s selected
binomial coefficients
Albert R Meyer,
April 21, 2010
lec 11W.6
Albert R Meyer,
April 21, 2010
lec 11W.7
‹#›
The Binomial Formula
Albert R Meyer,
April 21, 2010
The Binomial Formula
Albert R Meyer,
lec 11W.8
multinomial coefficients
(
in the expansion of
X 1 +X 2 +...+ X k
(X1+X2+X3+…+Xk)n ?
r ++r
1
k
April 21, 2010
lec 11W.9
The Multinomial Formula
What is the coefficient of
Albert R Meyer,
April 21, 2010
lec 11W.17
)
n
=
n
r1,r2,...,rk
=n Albert R Meyer,
r r r
r
X 11 X 22 X 33 ...X kk
April 21, 2010
lec 11W.18
multinomial coefficients
binomial a special case:
n n
=
k k, n - k More next lecture
about counting with
polynomials and series
Image by MIT OpenCourseWare.
Albert R Meyer,
April 21, 2010
Albert R Meyer,
April 21, 2010
lec 11W.20
‹#›
Pascal’s Identity
Preceding slides adapted from:
• Great Theoretical Ideas In Computer Science
Carnegie Mellon Univ., CS 15-251, Spring 2004
Lecture 10
Feb 12, 2004 by Steven Rudich
Algebraic Proof : routine, using
• Applied Combinatorics, by Alan Tucker
Albert R Meyer,
April 21, 2010
Albert R Meyer,
lec 11W.21
Combinatorial Proof
April 21, 2010
lec 11W.22
Combinatorial Proof
classify subsets of {1,…,n}
classify subsets of {1,…,n}
n n - 1
k = k # size k subsets =
# size k subsets with 1
+ # size k subsets without 1
n - 1
+
k - 1
# size k
subsets
Albert R Meyer,
April 21, 2010
Combinatorial Proof
n n - 1 n - 1 k = k + k - 1
# size k
subsets
# size k
subsets
without 1
April 21, 2010
lec 11W.24
classify subsets of {1,…,n}
QED
n n - 1 n - 1 k = k + k - 1
Albert R Meyer,
April 21, 2010
Combinatorial Proof
classify subsets of {1,…,n}
# size k
subsets
Albert R Meyer,
lec 11W.23
lec 11W.25
# size k
subsets
without 1
Albert R Meyer,
April 21, 2010
# size k
subsets
with 1
lec 11W.26
‹#›
Combinatorial Proof, II
Combinatorial Proof, II
classify subsets of {1,…,n,1,….,n}
2
n 2n
i = n i=0 n
Albert R Meyer,
April 21, 2010
2n
RHS = n
# size n
subsets
lec 11W.27
Combinatorial Proof, II
n
LHS = i=0 i n
LHS =
n
n
i
i=0
n n = i
n
i
i=0 April 21, 2010
April 21, 2010
lec 11W.29
Albert R Meyer,
n n
i
April 21, 2010
Combinatorial Proof, II
Combinatorial Proof, II
LHS =
n
n
i
i=0
LHS =
n
n
i
i=0
n n - i
April 21, 2010
lec 11W.30
n n - i
# size i
# size n-i
red subsets black subsets
# size i
red subsets
Albert R Meyer,
lec 11W.28
Combinatorial Proof, II
2
n
Albert R Meyer,
Albert R Meyer,
lec 11W.30
Albert R Meyer,
April 21, 2010
lec 11W.31
‹#›
Combinatorial Proof, II
n
i=0
n
i
Combinatorial Proof, II
n n - i
Therefore
LHS = # size n subsets = RHS
# size i
# size n-i
red subsets black subsets
So LHS = # size n subsets
of {1,…,n,1,….,n} by Sum Rule
2
n 2n
i = n i=0 n
QED
Albert R Meyer,
April 21, 2010
lec 11W.32
Albert R Meyer,
April 21, 2010
lec 11W.33
Team Problems
Problems
13
Albert R Meyer,
April 21, 2010
lec 11W.34
‹#›
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6.042J / 18.062J Mathematics for Computer Science
Spring 2010
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