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CSE 21: Notes 4
October 7, 2009
Binomial Coefficients
Recall that
n k
n! k!(n−k)!
which is the number of k-subsets of an n-set. The following are some properties
of binomial coefficients.
P1: (
n
k
n
k + 1
n + 1
k + 1
P2:
∑n
k=
k
m
n + 1
m + 1
Sketch of proof. Proof by induction:
n∑+
k=
k
m
∑^ n
k=
k
m
n + 1
m
n + 1
m + 1
n + 1
m
n + 1
m + 1
. (by P1)
Application of Binomial Coefficients
- Prove that ∑n
k=
k
2
n(n + 1)(2n + 1)
6
Sketch of proof.
∑^ n
k=
k
2
∑^ n
k=
k
2 − k + k
∑^ n
k=
k
2
k
1
∑^ n
k=
k
2
∑^ n
k=
k
1
n + 1
3
n + 1
2
(by P2)
n(n + 1)(2n + 1)
6
- Prove that ∑n
k=
n − k
k
= Fn+1,
where Fn is the nth Fibonacci number.
Sketch of proof. Proof by induction:
Fn+1 =
n
0
n − 1
1
n − 2
2
n − 3
3
n − 4
4
Fn+2 =
n + 1
0
n
1
n − 1
2
n − 2
3
n − 3
4
n − 4
5
q q q q q q q
Fn+2 =
n + 2
0
n + 1
1
n
2
n − 1
3
n − 2
4
n − 3
5
- (“Balls and Bins”) How many ways are there to throw n distinct balls into k distinct boxes if
a. No restrictions.
k
n .
b. There are ni balls in ith box for 1 ≤ i ≤ k.
There are n!
n 1 !n 2! · · · nk!
n
n 1
n − n 1
n 2
n − n 1 − n 2
n 3
nk
nk
ways.
c. There are ni balls in one of the boxes for 1 ≤ i ≤ k, i.e. the if order doesn’t matter.
There are 1
k!
n!
n 1 !n 2! · · · nk!
k!
n
n 1
n − n 1
n 2
n − n 1 − n 2
n 3
nk
nk
ways.
- How may ways are there to partition 12 students into 4 teams such that each team has 4 students if
a. No restrictions.
b. There are 6 girls and 6 boys among the 12 students and each team has two 2 girls.
The number of ways to partition 6 boys into 3 teams where each team has 2 boys is (here, the
order of team doesn’t matter) 6!
2!2!2!
Then, the number of ways to partition 6 girls into 3 teams where each team has had 2 boys already
is 6!
2!2!2!
Thus, there are in total 15 × 90 = 1350 ways.
c. Each team has at least one girl.
Hint: Count the complement.
- (“Bars and Stars”) How many ways to throw 5 marbles into 3 boxes?
Consider arranging 5 marbles and 2 ‘|’ ’s in a line, e.g.
marble − marble − | − marble − marble − | − marble
which can be viewed to put 2 marbles into the 1st box, 2 marbles into the 2nd box, and 1 marble into
the 3rd box. Since there are
2
ways to arrange 5 marbles and 2 ‘|’ ’s, there are
2
ways to throw 5
marbles into 3 boxes.
In general, the number of ways to throw n marbles into k boxes is
(n+k− 1
k− 1
- How many ways to give 10 cookies to 5 students if
