Try It Now

Site: Saylor Academy
Course: CS202: Discrete Structures
Book: Try It Now
Printed by: Guest user
Date: Thursday, February 22, 2024, 1:43 PM

Description

Work these exercises to see how well you understand this material.

Table of contents

Exercises

  1. If a raffle has three different prizes and there are 1,000 raffle tickets sold, how many different ways can the prizes be distributed?

  2. How many eight-letter words can be formed from the 26 letters in the alphabet? Even without concerning ourselves about whether the words make sense, there are two interpretations of this problem. Answer both.

  3. The state finals of a high school track meet involves fifteen schools. How many ways can these schools be listed in the program?

  4. All 15 players on the Tall U. basketball team are capable of playing any position.
    1. How many ways can the coach at Tall U. fill the five starting posi- tions in a game?
    2. What is the answer if the center must be one of two players?

  5. The president of the Math and Computer Club would like to arrange a meeting with six attendees, the president included. There will be three computer science majors and three math majors at the meeting. How many ways can the six people be seated at a circular table if the president does not want people with the same majors to sit next to one other?

  6. Let A = {1, 2, 3, 4}. Determine the cardinality of
    1. {(a1, a2) | a1 ≠ a2}
    2. What is the answer to the previous part if |A| = n
    3. If |A| = n, determine the number of m-tuples in A, m ≤ n, where each coordinate is different from the other coordinates.

 


Source: Al Doerr and Ken Levasseur, http://faculty.uml.edu/klevasseur/ads-latex/ads.pdf
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

Solutions

  1. Answer: P (1000, 3)

  2. Answer: With repetition: 268 ≈ 2.0883 × 1011. Without repetition: P (26, 8) ≈ 6.2991 × 1010

  3. Answer: 15!

  4. Answer:
    1. P (15, 5) = 360360
    2. 2 · 14 · 13 · 12 · 11 = 48048

  5. Answer: 2 · P (3, 3) = 12

  6. Answer:
    1. P (4, 2) = 12
    2. P (n; 2) = n · (n − 1)
    3. Case 1: m > n. Since the coordinates must be different, this case is impossible.
      Case 2: m ⩽ n.P (n ; m).