CS173: Discrete Mathematical Structures
Spring 2006
Homework #8
Due 03/19/06, 8am
1. Use Pigeonhole Principle to prove followings:
a. Suppose each point of the plane is color green or yellow. Prove that some
line segment has ends in the same color.
b. Show that if we randomly choose 17 points within an equilateral triangle
(with its side equal to 1), there must be a pair of points at a distance less
than 0.25.
c. If 9 balls are to be placed in a row of 12 bins (one bin can hold only one
ball), there must be a consecutive set of 3 bins that are filled with balls.
2. Binomial Theorem:
a. Find the 6th term in the expansion of 10
3
1
()x
x
−. Then find the coefficient
of 14
1
x
in this expansion.
b. Use the binomial theorem twice to expand 3
()
xy
z++ .
c. Use the binomial theorem to prove that
2
(,0) (,1)2 (,2)2 (,)2 3
nn
Cn Cn Cn Cnn++ +⋅⋅⋅+ =
3. Consider a computer network with 60 switching nodes. The network is designed to
withstand the failure of any two nodes.
a. In how many ways can one or two nodes fail?
b. In how many ways can a failure of network occur?
c. If one node has failed, in how many ways can seven nodes be selected
without encountering the failed node?
d. If two nodes have failed, in how many ways can seven nodes be selected
to include exactly one failed node?
4. Consider a 5-card hand from a standard 52-card deck
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