MAT 4830Numerical Analysis
Binomial Coefficients and Combinatorial Identities
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Goals
Binomial Theorem Binomial Coefficients Combinatorial Identities Review shifting indices Review Induction
Take Home Exam
Need Binomial Coefficients for the second problem.
Need Binomial Theorem for a few parts of the second problem.
Binomial Expansion
2 2 2
3 3 2 2 3
2
3 3
? n
a b a ab b
a b a a b ab b
a b n Z
Binomial Theorem
0
!Binomial Coefficients:
,
,
! !
nn n r r
r
a b R
na b
n n
r
a br
n
Z
r r
n
Useful Formulas for Binomial Coefficients
, 0
1. 1 and 10
2.1
3.
Easy to check.
n Z r n
n n
n
nn
n n
n r r
Pascal’s Identity1
, 0 11 1
1, 1
1
n n nr n
r r r
OR
n n nr n
r r r
1, 0 1
1 1
1, 1
1
n n nr n
r r r
OR
n n nr n
r r r
Pascal’s Identity
Proof: Analysis
Binomial Theorem
0
, n
n n r r
r
na b a b n Z
r
Combinatorial Proof: Analysis
Example 1
Find the coefficient of in the expansion of .
Example 2
0
2 , n
n
r
nn Z
r
Proof: Analysis
0
,n
n n r r
r
na b a b
r
Example 3 (a)1
Given a fixed integer 0, , 1
n
i r
i nr n r
r r
Proof:1. Induction2. Can be done without induction, but need to take care special cases.
Analysis
1, 0 1
1 1
n n nr n
r r r
Example 3 (b)
1
Use (a) to find a formula for 1 2n
i
i n
Solution: Analysis
1
1
?
n
i r
i
r
r
n
r
Binomial Theorem
0
, n
n n r r
r
na b a b n Z
r
Induction Proof:Need some preparations
Analysis
Binomial Theorem
Proof: Analysis
11. , 0 1
1 1
2. 1 an
3. Index Shifting*
d 1, 0
n n nr n
r r r
m mm Z
m
0
, n
n n r r
r
na b a b n Z
r
3. Recall: Index Shifting for Summations (Use this if…)
1
1
1
1
( ) ( 1)
( 1)
n n
r m r m
n
r m
f r f r
f r
Index Shifting
Sigma representation of a summation is not unique
22226
2
22
22224
0
22
22225
1
22
543211
543211
54321
i
i
i
i
i
i
Index Shifting Rules
6
2
24
0
25
1
2 11iii
iii
Index Shifting Rules
6
2
24
0
25
1
2 11iii
iii
decrease the index by 1
increase the i in the summation by 1
Index Shifting Rules
6
2
24
0
25
1
2 11iii
iii
increase the index by 1
decrease the i in the summation by 1