Discrete Mathematics Answers

Find the inverse of 55 modulo 7 by using extended Euclidean Algorithm

step by step solution

The English alphabet contains 21 consonants and 5 vowels. How many strings of five lowercase letters can be formed using the following constraints? Give two answers for each of the following - one where repetition is allowed in the string and one where repetition is not allowed.

(a) Only one vowel (placed anywhere)

(b) Maximum two consonants (placed anywhere)

(c) Starts with x, y or z

State and prove Pascal’s identity using the formula for "{n \\choose k}"

Consider all strings of length 12, consisting of all uppercase letters. Letters may be repeated. Please do not simplify your answers.

(a) How many such strings are there?

(b) How many such strings contain the word ”SCOOBY”?

(c) How many such strings contain neither the word ”SCOOBY” nor the word ”DAPHNE”?

How many solutions in non-negative integers are there to the equation x₁ + x₂+ x₃ + x₄ = 19

Answer the following:

(a) You have a standard 52 card deck. If you draw two cards at random, what is the probability that they are both hearts?

(b) You reshuffle your cards into the deck, but unfortunately now a mischievous dog (a golden retriever named Bubbles), decides to eat one of the cards! Assuming that the missing card is a diamond, now when you draw two cards at random what is the probability that they are both hearts?

Using Binomial Theorem, give the closed form expression for: "\\textstyle\\sum_{k=0}^n""{n \\choose k}"3ⁿ · 2^k

in a class of final year student in a school, 350 offer chemistry and 200 offer physics. if 90 student offer neither chemistry nor physics;

i: how many students are offering both subject.

ii: how many students are offering only one subject.

iii: how many students are offering at least one subject

For this arguments, explain which rules of inference are used for each step. Same body in this class enjoys whole watching. Every person who enjoys whole watching cares about ocean pollution. Therefore, there is a person in this class who cares about ocean pollution.

Consider the sets A and B, where A = {3, |B|} and B = {1, |A|, |B|}. What are the sets?