Download Assignment 5: Combinatorics

Math 240: Discrete Structures I
Due: Thursday 20th November
Assignment 5: Combinatorics
1. Recurrence Equations.
(a) Solve the recurrence equation
g(n) = 6g(n − 1) − 7g(n − 2) ∀n ≥ 2; g(0) = 1, g(1) = 3
(b) Let fn be the number of sequences with alphabet {0, 1, 2} of length n
that do not contain two consecutive even numbers. Find and solve the
recurrence relation for fn .
2. Inclusion-Exclusion. Two letters need to be delivered to each of n houses. How
many ways can a postman deliver two letters to each house such that each house
receives at least one incorrect letter?
3. Inclusion-Exclusion II.
(a) Use inclusion-exclusion to count the number of positive integers up to and
including 50 that are square-free.1
(b) Consider 100 students, each taking at least one of the courses: art, biology
and computing. Let 20 students take both art and biology, 31 students take
both art and computing, and 24 students take both biology and computing.
Furthermore, 85 students take at least one of art and biology, 78 students
take at least one of art and computing, and 84 students take at least one of
biology and computing. How many students are taking all three courses?
4. Pigeon-Hole Principle. Consider the 2D grid with integer coordinates.2 Prove
that if we take five points on the grid then there exist two of the points whose
average is also a point on the grid.
5. Pigeon-Hole Principle. Consider a set {x1 , x2 , . . . , xn } of n integers. Prove that
there exist indices s and t (where 1 ≤ s ≤ t ≤ n) such that
t
X
xi ≡ 0 mod n
i=s
1
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A number is square free if it is not divisible by the square of an integer larger than 1.
That is, a point (x, y) lies on the grid if and only if both x and y are integral.
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6. Multinomial Coefficients.
P
(a) Let n = ri=1 ki , where each ki is positive. Show that
n
k1 , k2 , . . . , kr
=
r X
i=1
n−1
k1 , . . . , ki−1 , ki − 1, ki+1 , . . . , kr
P
(b) Let p be a prime number with p = ri=1 ki , where each ki is non-negative
and at least two of them are positive. Prove that
p
k1 , k2 , . . . , kr
is divisible by p.
(c) Let h(n, r) be the number of surjections from [n] to [r]. Give a formula for
h(n, r) in terms of multinomial coefficients.
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