ArsDigita University
Month 2:
Discrete Mathematics  Professor Shai Simonson
Problem Set 6 – Combinatorics and Discrete Probability
1.
Assume someone is throwing three dice of different colors.
a.
How many ways are there to roll the dice?
b.
Make a chart showing how many of these rolls, have
i
as the second highest
and
j
as the highest value, for
1
≤
i
≤
j
≤
6
.
(Feel free to write a program to
do it, if that will be faster).
c.
Sum up the columns and rows of your chart and state an interesting theorem
about the number of rolls whose second highest die value is
m.
d.
A generalization of this theorem for
n
dice, where
n
is an odd number, is that
the number of rolls of
n
dice whose median value die equals
m
is the same as
the number of rolls of
n
dice whose median value die equals
7

m.
Prove this
theorem.
2.
In the game of Risk, one player, the attacker, throws three dice.
The defender
chooses to roll either one die or two dice.
If the defender rolls two dice, then the two
highest rolls of the attacker are compared with the rolls of the defender.
The higher is
compared against the higher and the lower against the lower, with a tie going to the
defender.
In every battle with two dice, the defender can win two, lose two or split.
If the defender rolls just one die, then only the highest roll of the attacker is compared
to it, and again a tie goes to the defender.
Here the defender can win one or lose one.
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 Spring '09
 Randomness, Dice, 2.5%, ArsDigita University, Shai Simonson

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