Discrete Mathematics: Combinatorics, Probability, and Sets Assignment

This document presents solutions to a discrete mathematics assignment focusing on fundamental concepts of sets, combinatorics, and probability. The assignment includes problems on calculating dessert choices with constraints, determining password possibilities with uppercase and lowercase letters, and analyzing binary strings of length 8. Additionally, it addresses problems related to the Konami code, calculating possible character sequences under specific conditions. The solutions demonstrate the application of combinatorial principles to solve various counting problems, including permutations and combinations, and provide a comprehensive approach to each question, making it a valuable resource for students studying discrete mathematics. The assignment covers a range of topics, providing a solid foundation in these core mathematical concepts.

ASSIGNMENT

Discrete Mathematics: Sets, Combinatorics, and Probability
2. A frozen yogurt shop allows you to choose one flavor (vanilla, strawberry, lemon,
cherry, or peach),
one topping (chocolate shavings, crushed toffee, or crushed peanut brittle), and one
condiment (whipped
cream or shredded coconut). How many dessert choices do you have if you are allergic
to strawberries and chocolate?
Sol.
Flavors available :
1. Vanilla
2. Strawberry
3. Lemon
4. Cherry
5. Peach
Toppings available:
a) chocolate Shavings
b) Crushed toffee
c) crushed peanut brittle
Condiment available :
i. whipped cream
ii. shredded coconut
Since, we can not choose 2 flavors, i.e., Strawberry and chocolate, number of choices
left for flavours = 5-2=3
Number of choices available in toppings = 3
Number of choices available in condiment = 2
total choices available =
3 * 3 * 2 = 18
6. A user’s password to access a computer system consists of 3 letters followed by 2
digits. How many passwords are possible if uppercase and lowercase letters can be
distinguished?
Sol.
Total letters available = 26 upper case + 26 lower case = 52
52 52 52 10 10
Total choices = 52*52*52*10*10 = 1,40,60,800
Exercises 33–42 concern the set of binary strings of length 8 (each character is either
the digit 0 or the digit 1).
34. How many begin and end with 0?
0 * * * * * * *
Choices = 1*2*2*2*2*2*2*2 = 128
36. How many have 1 as the second digit?
* 1 * * * * * *
Choices = 2*1*2*2*2*2*2*2*2 = 128
40. How many are palindromes? (See Exercise 11.)

* * * *
Choices = 2*2*2*2*1*1*1*1=16
42. How many contain two or more 0s?
0 0 * * * * * *
Choices = 1*1*2*2*2*2*2*2 =64
Exercises 55–58 refer to the Konami code, a sequence of 10 keystrokes using 6 different
characters:
c c T T d S d S B A. This code was inadvertently left by the developer in the first release
of a video game;
when entered on the video game console as the title screen is open, it surreptitiously
adds assets to the player’s
on-screen avatar. This code or versions of it have been retained in many video games.
56. How many 10-character codes can be created if the BA sequence must be the last 2
characters?
B A
The two characters B and A are fixed.
This means number of characters left = 6-2 = 4
Choices left = 4*4*4*4*4*4*4*4*1*1=65,536
58. How many 10-character codes can be created if only the arrow characters are used?
There are only 4 possible arrow characters.
Choices left = 4*4*4*4*4*4*4*4*4*4=1,048,576