Dr. Timo de Wolff
Institute of Mathematics
www.math.tamu.edu/~dewolff/Spring16/math302.html
MATH 302 – Discrete Mathematics – Section 501
Homework 8
Spring 2016
Due: Wednesday, April 13th, 2016, 4:10 pm.
When you hand in your homework, do not forget to add your name and your UIN.
Your solution for all the following problems should not only be the correct number, but
also contain an argumentation how you determined the correct number.
Exercise 1. Assume a dice is rolled five times. How many combinations exist, which
contain in total at least two 1’s?
Exercise 2. How many possibilities exist to place 8 rooks on a chessboard such that no
rook can capture another one with one move? Explain your answer.
Hint: A chessboard consists of 64 squares arranged in an 8 times 8 grid. A rook can
capture every piece in one move that is in a straight vertical or horizontal line from its
current position.
Exercise 3. In our garden live a couple of lizards, which can change their color. I made
the following observations:
1. There are 5 (distinguishable) lizards: 2 live under the house, 2 in the old tree and 1
in the shed. They change their color only between gray, brown and green.
2. The 2 lizards under the house are always gray.
3. The 2 lizards in the old tree have an arbitrary color, but they never have the same
color.
4. The lizard in the shed always has the same color as one of the other lizards.
How many combinations of colors can the 5 lizards in our garden have?
Exercise 4. My aunt has seven black cats she alone can distinguish: Zeus, Poseidon,
Athena, Aphrodite, Ares, Hera, and Apollo . The cats have seven different favorite sorts
of food, seven different favorite places to sleep, and seven favorite toys.
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1. The cats live in harmony: They exchange their favorite food, places to sleep, and
favorite toys among each other every week. However, every sort of food, every place,
and every toy always belongs to exactly one cat. How many of such combinations
exist in total?
2. Recently, Athena did not feel well and did not like to eat anything. Hence, Aphrodite
and Apollo decided to stay with Athena instead of picking an own place to sleep.
Aphrodite brought her favorite toy with her and also picked one for Athena, while
Apollo was so worried that he was not interested in toys that week. During that time
when Athena did not eat, Aphrodite and Apollo did not have an own place to sleep,
and Apollo did not have a favorite toy: How many possible combinations existed?
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