Assignment 10 (More Counting)

1. According to Sonoma State University policies (stated on the SSUmpage to change one's passwords after logging in),

   Passwords must be:

   • At least 10 and no more than 28 characters long (Important!)
   • A combination of letters, numbers and special characters, containing at least three of the following character types:
     • Lowercase alphabetic character (a-z)
     • Uppercase alphabetic character (A-Z)
     • Special character (punctuation, spaces, *, %, $, etc.)
     • Number (0-9)
   • Don't use your name, birthday, or social security number!

   Altogether these rules can be reasonably complex to analyze, so this problem makes some simplifications.

   • Passwords will be exactly 10 characters long.
   • Only the following 7 characters will be considered "special": !@#$%&*
   • The only names to be considered will be "Henry" and "Walker". Also, neither birthdays nor social security numbers will be checked.

   For the following, count the number of passwords with the specified constraints, if any.

   1. How many 10-character strings (i.e., potential passwords) can be formed from uppercase and/or lowercase letters, special characters, and digits without constraints (i.e., ignoring any requirement regarding one type of character or another).

   2. How many 10-character strings contain at least one special character and at least one digit.

   3. How many 10-character strings (i.e., potential passwords) contain either the name "Henry" or the name "Walker", together with at least one special character and at least one digit character. (Additional uppercase or lower case letters also are allowed to yield 10 characters altogether.)

   Notes:

   • Passwords of exactly 10 characters in length cannot contain both "Henry" and "Walker".
   • Either "Henry" or "Walker" meets requirements for at least one uppercase letter and at least one lowercase letter.
   • When counting 10-character strings, be sure to count the string "HenryHenry" only once.

2. A software design class has 24 students enrolled, and classwork will be organized around 6 different components of an overall software project. Throughout the semester, students will be divided into teams of 4, with each team developing one of the components. How many ways can the students be organized into teams? (Here, each team has four students, and each team has its own project. Membership of each team is disjoint from the others.) As part of your answer, explain your reasoning that led to your answer.

3. A purchasing agent in an electronics store plans to place an order for 50 laptop computers. In more detail the order will include at least 2 models of HP computers, 3 models of Lenovo machines, 2 models of Acer laptops, 3 models of Dell machines, and 5 models of Macintosh laptops. In considering a specific model (e.g., 1 of the 5 Macintosh computers), all of the ordered laptops will be identical in all respects.

   1. Assuming no constraints on the number of machines to be ordered for each model, how many ways are there to place the order? As part of your answer, explain your thought process that led to your computation.

   2. How many ways are there to place the order, assuming the purchasing agent must buy at least 3 laptops of each model. Again, explain how you reached your conclusion.

created October 25, 2025 revised October-November, 2025
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.

Copyright © 2011-2021 by Henry M. Walker. This page and other materials developed for this course are under development and licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.