CS173: Discrete Mathematical Structures Fall 2006
Homework 10
Due: 8am, November 21, 2006
November 14, 2006 Cinda Heeren
Problem 1
(10 points) How many solutions are there to the equation
x1+x2+x3+x4+x5+x6= 29
where xi, i = 1,2,3,4,5,6,is a nonnegative integer such that
a) xi>1 for i= 1,2,3,4,5,6?
b) x1≥1, x2≥2, x3≥3, x4≥4, x5≥5,and x6≥6?
c) x1≤5?
d) x1<8 and x2>8?
Problem 2
a) (3 points) How many different strings can be made from the letters in MI SS IS SI P P I,
using all the letters?
b) (4 points) How many different strings can be made from the letters in MI SS IS SI P P I,
using all the letters and having no consecutive Ss?
c) (3 points) How many ways are there to travel in xyzw space from the origin (0,0,0,0) to the
point (4,3,5,4) by taking steps one unit in the positive x, positive y, positive z, or positive
wdirection?
Problem 3
(10 points) What is the probability of these events when we randomly select a permutation of the
26 lowercase letters of the English alphabet?
a) ais the first letter of the permutation and zis the last letter.
b) yand zare not next to each other in the permutation. (Hint: it might be easier to first think
of the probability of yand zbeing next to each other.)
c) aand zare separated by at least 23 letters in the permutation. (Hint: list all possible
configurations and then figure out how many permutations each configuration can have. For
example, a♦24zis a possible configuration)
d) zis somewhere in between aand bin the permutation.
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