Homework 1#

Due: Thursday, August 27, 2026 23:58 Submit: Gradescope, through Blackboard’s assignment area.

Please write your homework by hand. Start each exercise on a new page, scan your work to a legible PDF, and clearly mark the pages associated with every exercise.

1. (Using every letter)#

Points: 10

How many strings of length 10 over the alphabet contain each of the three letters at least once?

Need a free hint?#

Count all strings, then subtract the forbidden strings. A forbidden string uses exactly one or exactly two different letters. How many use only , only , or only ? For each of the pairs , , and , how many strings use both letters in the pair and no third letter?

2. (Forming a committee)#

Points: 10

A company has 20 employees in Marketing and 10 employees in Research and Development (R&D). It must form a committee with the following five positions:

  • one chair;

  • one vice-chair; and

  • three ordinary members, whose positions are not distinguished from one another.

No employee may hold more than one position. The chair and vice-chair must come from different departments; the ordinary members may come from either department. How many committees are possible?

Need a free hint?#

Split into cases according to which department provides the chair and which provides the vice-chair. Choose the officers first, then the three ordinary members; remember that the ordinary-member positions are not distinguished.

3. (Digits on a pentagon, without repetition)#

Points: 12

Place five distinct digits chosen from at the vertices of a regular pentagon. Two patterns are considered the same when one can be obtained from the other by rotating the pentagon. Reflections are not considered the same.

The examples below illustrate this convention. The labels name the fixed vertex positions , beginning at the top vertex and proceeding clockwise.

Pentagon with digits 1, 2, 3, 4, 5 clockwise from the top vertex.

Figure 1: Pentagon with digits 1, 2, 3, 4, 5 clockwise from the top vertex.#

A rotation of Pattern 1, with digits 5, 1, 2, 3, 4 clockwise from the top vertex.

Figure 2: A rotation of Pattern 1.#

Pentagon with digits 1, 2, 3, 5, 4 clockwise from the top vertex.

Figure 3: A nonrotation of Pattern 1 using digits 1, 2, 3, 5, 4.#

Pentagon with digits 1, 2, 4, 6, 8 clockwise from the top vertex.

Figure 4: A pattern using digits 1, 2, 4, 6, 8.#

Patterns 1 and 2 are considered the same because a rotation carries one to the other. No other pair among the four displayed patterns is considered the same. How many different patterns are there?

Need a free hint?#

First treat as fixed, distinguishable positions and count the assignments of five distinct digits. Next determine how many labeled arrangements represent each pattern up to rotation, explaining why the ``no repetition’’ condition makes this number the same for every pattern. Finally apply the division rule and simplify to a numerical answer.

4. (Why division by five can fail)#

Points: 8

Now change Exercise 3 by allowing digits to repeat. Thus each of the five vertices may receive any digit from , while patterns that differ only by a rotation are still considered the same.

A tempting argument is:

There are assignments to five fixed vertices, and every pattern is represented five times by rotation, so the answer is .

Explain carefully why this argument is incorrect. You are not being asked to find the correct total.

Need a free hint?#

Explain why is the correct number of assignments when the vertices are fixed and labeled. Then list the distinct rotations represented by each of the clockwise strings 00000, 00001, and 01234, beginning at vertex , and use them to identify which requirement of the division rule fails. Focus on whether every pattern is counted the same number of times.