MC215 – MATHEMATICAL REASONING & DISCRETE STRUCTURES – 4/24/14 – HOMEWORK #6
Due Wednesday, 4/30/14 by 5 PM -- You may work alone or in teams of 2
OPTIONAL TEAM OF 2 ASSIGNMENT: As with HW #4, if you wish (and I encourage you to do so), you
may work together with one partner and submit a single assignment with both names on it. If you'd like
to do this but need help finding a partner, let me know as soon as possible. If you do work with a
partner, please do not collaborate with anyone else. If you decide to work alone, you may as usual
consult with other students who are also working alone.
Be sure to justify all answers on this assignment. Give numeric answers both as an algebraic
combination indicating how they were obtained, and also as an actual number. For example, the
number 3-letter strings using 3 distinct lowercase letters is: 26 × 25 × 24 = 15,600. For probabilities,
give both the fraction and the decimal to 2 or 3 places. For example,
×
×
= 0.888.
1. Skidmore College has a web page1 giving the rules for an acceptable password. Here is a simplified
version of what it says on that page:
i.
For your security, Skidmore requires that your password be between 6 and 8 characters
in length.
ii. The only characters that may be used are upper and lower case letters, numeric
characters, and characters from this set of 19 special characters:
{}!#”$%^&,()_+-=:;?
iii. Each password must contain at least one letter (upper or lower case) and at least one
numeric character.
a. Let P be the set of passwords that satisfy Rules (i) and (ii), i.e., length 6-8, using only
permissible characters (26 uppercase letters + 26 lowercase letters + 10 numeric characters+
19 special characters = 81 permissible characters). Elements of P do not have to satisfy Rule
(iii). How many passwords are in P?
b. Let L be the set of passwords in P that contain at least one letter.
Let N be the set of passwords in P that contain at least one numeric character.
Let S be the set of passwords in P that contain only special characters.
Compute the number of passwords in each of these sets, i.e., compute ||, ||, and |S|. Hint:
For L and N, first compute the size of the complement, and subtract the total from |P| to get
the size of the original set. You will also need the sizes of the complements for part (d).
c. The set ∩ is the set of passwords satisfying Rules (i), (ii), and (iii). Prove that | ∩ | =
|| − − + ||. Hint: First apply DeMorgan’s Law to the set ∩ , and then apply the
Inclusion-Exclusion Principle to compute the size of the resulting union of two sets.
d. Using the results of parts (a)-(c), how many passwords satisfy Rules (i)-(iii)?
1
https://www.skidmore.edu/it/policies/passwords.php
2. Suppose we take a blank piece of paper, 8½” by 11”, and we mark five distinct points on it in random
locations.
a. Prove that there must be two points whose distance from each other is strictly less than 7”.
Hint: Divide the piece of paper into 4 equal size rectangles, and use the Pigeonhole Principle.
You’ll also need the Pythagorean Theorem: If a right triangle has hypotenuse of length c and
the other two sides have lengths a and b, then c2 = a2 + b2.
b. How many points would we have to mark to guarantee that there are three points that are
each strictly less than 7” from each of the other two? Justify your answer, and also draw a
diagram, with one fewer point than your answer, that demonstrates that your answer is the
smallest number that guarantees three such points. Explain this answer as well.
1) Today I bought a bag of M&Ms (for professional reasons only), and I
counted the number of each color; see included photo. You can check that
there were (note past tense) 9 blue, 10 brown, 9 green, 12 orange,
6 red, and 9 yellow M&Ms. Suppose I put them all back in the bag (as if!)
and mix them all up. If I then reach in and pick one out, there are six
possible outcomes: blue, brown, green, orange, red, and yellow. The set of
these outcomes will be the sample space for this problem.
a) Define an appropriate probability distribution on S, using the numbers of each color given
above. For each œ, say what () is, both as a fraction of two integers, and as a decimal
between 0 and 1.
b) The colors yellow and green are “Skidmore Colors,” while the colors red, blue, and green are
“Primary Colors.” What is the probability that an M&M picked at random from a full bag is a
Skidmore color or a primary color?
c) Suppose I pick two M&Ms from a full bag. What is the probability that I pick both a brown
M&M and an orange M&M if:
i) I place the first M&M back before taking the second, and
ii) I do not place the first M&M back before taking the second?
Hint: In each part it might be helpful to think of this as 4 events: #$ =the first
M&M is brown, %$ =the first M&M is orange, #
=the second M&M is brown,
%
=the second M&M is orange. Then the event whose probability we want to
compute is (#$ ∩ %
) ∪ (%$ ∩ #
), which is a union of two mutually exclusive
events, so you can just add up the probability of each one to get the total
probability.