Math 274 Combinatorics Spring 2015
Solutions to Problems for 12th February
2.16. Find the number of solutions of x1 +x2 +.+xk = n, where k 2, x1 and x2 are arbitrary nonnegative
integers, and x3 , x4 , ., xk are nonnegative even integers.
Solution: The ge
Math 274
Combinatorics
Spring 2015
Combinatorial e.g.fs: Have A = set of objects, w = weight function, ak objects of weight k,
dene
a2 x 2
ak xk
a3 x3
xw()
A(x) = a0 + a1 x +
+
+ . =
=
2!
3!
k!
w()!
k=0
A
Examples: (1) E.g.f for r-permutations of an n-set
Math 274 Combinatorics Spring 2015
Solutions to Problems for 15th January
1.2. (a) In the Canadian province of Ontario licence plates used to have one of two forms: LLL-DDD or
DDD-LLL, where L represents a position occupied by a letter, and D a position o
Math 274
Combinatorics
Spring 2015
1. PERMUTATIONS AND COMBINATIONS
Sum and product rules
Sum Rule: Suppose we have k mutually disjoint sets of objects S1 , S2 , . . ., Sk . Then the total
number of objects is
|S1 | + |S2 | + . . . + |Sk |.
Note: An objec
Math 274
Combinatorics
Spring 2015
Many nonstring problems can be turned into string problems.
Example: Suppose we have two red hats and unlimited numbers of green and blue hats. In how
many ways can we give n distinct people a hat each?
Solution: Model b
Math 274
Combinatorics
Spring 2015
General ordinary generating functions
Will look at another sort of generating functions, exponential g.f.s, later.
Generating function of a sequence: The (ordinary) generating function (o.g.f. or g.f.) of a
sequence (a0
Math 274
Combinatorics
Spring 2015
Example: What is the probability of a ush in poker? (5-card, no drawing, count as a ush every
hand with all cards of same suit).
52
Solution: Total number of hands is
.
5
13
To construct a ush: pick a suit (4 ways) then
Math 274 Combinatorics Spring 2015
Solutions to Problems for 29th January
1.18.
n
n+1
n+2
n+r
+
+
+ +
0
1
2
r
n+r+1
r
=
Solution: Several proofs.
(1) By induction:
n+r+1
r
=
=
n+r
n+r
+
r
r1
n+r
n+r1
n+r1
+
+
r
r1
r2
= .
=
=
n+r
n+r1
n+1
n+1
+
+ .+
+
r
r1
Math 274 Combinatorics Spring 2015
Solutions to Problems for 22nd January
1.5. A committee is to be chosen from 7 men and 5 women. How many ways are there to form it if:
(a) it has 5 people, 3 men and 2 women?
(b) it can be any positive size but must have
Math 274
Combinatorics
Spring 2015
Subset problems
Examples: (1) Find the number of (a) 5-subsets, (b) k-subsets, of Nn with no two consecutive
integers.
Solution: To count subsets, we are going to count dierence vectors.
(a) Consider S = cfw_a1 , a2 , .
Math 274 Combinatorics Spring 2015
Solutions to Problems for 19th February
2.20. Find the number of 4-subsets cfw_a1 , a2 , a3 , a4 (a1 < a2 < a3 < a4 ) of N30 with ai + i ai+1 ai + 3i,
1 i 3.
Solution: [7] (1) Set up a dierence vector (a1 , d1 , d2 , d3
Math 274 Combinatorics Spring 2015
Solutions to Problems for 12th March
2.29. Consider the set of strings made from as, bs and cs so that there is at least one a, the number of bs
is not exactly one, and the number of cs is not exactly two. How many such
Math 274 Combinatorics Spring 2015
Solutions to Problems for 5th February
2.2. Show that the generating function for all odd subsets of Nn = cfw_1, 2, ., n (with weight = cardinality)
is
(1 + x)n (1 x)n
.
D(x) =
2
(Hint: use the Binomial Theorem twice.)
S
Math 274
Combinatorics
Spring 2015
Binomial identities: Equations involving relationships between binomial coecients.
n
n
n
n
(1)
+
+
+ . +
= 2n .
0
1
2
n
Proof 1: Use Corollary to Binomial Theorem, let x = 1: get
n
n 1
n 2
n n
+
1 +
1 + .
1
0
1
2
n
n
n
n