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At a family group meeting of 30 women, 17 are descended from George, 16 are descended from John, and 5 are not descended from George or John. How many of the 30 women are...
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The number of distinct simple graphs with up to three nodes is $15$ $10$ $7$ $9$
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Determine the domain of the function $f(x) = |x – 2|$
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A pennant is a sequence of numbers, each number being $1$ or $2$. An $n-$pennant is a sequence of numbers with sum equal to $n$. For example, $(1,1,2)$ is a $4-$pennant. ...
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Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at $(i,j)$ then it can move...
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Prove that $2n < (n + 1)!,$ for all $n \geq 3.$
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Let $A$ be a set of $n(>0)$ elements. Let $N_r$ be the number of binary relations on $A$ and let $N_f$ be the number of functions from $A$ to $A$ Give the expression for ...
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Simplify $(A\cup B)\cap (A\cup B')\cap (A - B)$ for a given non empty sets $A$ and $B$, where $(A\cap B) = \varnothing .$
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Sketch the graph of $f(x) = \frac{x^{3}-1}{x^{2}-1}$
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Let y in the form of $a + bi$, where $a$ and $b$ are real numbers, be the cubic roots of complex number $z^{20},$ where $z=\frac{2}{4 + 3i}.$ Find $a + b.$
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Suppose $x, y, z > 1$ are integers, let: $p(x,y)$ : $x$ is a factor of $y$ $q(x,y,z)$ : $z$ = $\text{GCD}(x,y)$ $r(x)$ : $x$ is prime. Check if the following argument is ...
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A deck consists of 52 playing cards which is well shuffled. Draw 6 cards. Find the probability that among the cards there will be a representative of all suits? can someo...
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For the composition table of a cyclic group shown below: \begin{array}{|c|c|c|c|c|} \hline \textbf{*} & \textbf{a}& \textbf{b} &\textbf{c} & \textbf{d}\\\hline \textbf{...
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How to solve this question?
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Let $P,S,R$ be three statements(propositions). Let $S$ be a sufficient condition for $P$, Let $R$ is a necessary condition for $P$ then which of the following is/are true...
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Coefficient of x^8 in ( (1-x^6)/(1-x) )^3.
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The $2^n$ vertices of a graph $G$ corresponds to all subsets of a set of size $n$, for $n \geq 6$. Two vertices of $G$ are adjacent if and only if the corresponding sets ...
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Let $G$ be an arbitrary graph with $n$ nodes and $k$ components. If a vertex is removed from $G$, the number of components in the resultant graph must necessarily lie dow...
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Consider the undirected graph $G$ defined as follows. The vertices of $G$ are bit strings of length $n$. We have an edge between vertex $u$ and vertex $v$ if and only if ...
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Which one of the following is TRUE for any simple connected undirected graph with more than $2$ vertices? No two vertices have the same degree. At least two vertices have...
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In how many ways can $15$ indistinguishable fish be placed into $5$ different ponds, so that each pond contains at least one fish? $1001$ $3876$ $775$ $200$
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How many strings of $5$ digits have the property that the sum of their digits is $7$? $66$ $330$ $495$ $99$
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How to determine for which m, n the complete bipartite graph $Km,n$ is planar? I am getting two answers from two sources:- A complete bipartite graph $Kmn$ is planar if a...
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What is the value of the given determinant ?
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How many substrings (of all lengths inclusive) can be formed from a character string of length $n$? Assume all characters to be distinct, prove your answer.
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Graph $G$ is obtained by adding vertex $s$ to $K_{3,4}$ and making $s$ adjacent to every vertex of $K_{3,4}$. The minimum number of colours required to edge-colour $G$ is...
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A deck of 5 cards (each carrying a distinct number from 1 to 5) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is the probability th...
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An unbiased coin is tossed repeatedly until the outcome of two successive tosses is the same. Assuming that the trials are independent, the expected number of tosses is $... 2 answers 29 For a complete graph with 10 vertices, The number of spanning trees is at least_____? 2 answers 30 Which of the following associative multiplication tables defined on the set$G = \{a, b, c, d\}$form a group? 2 answers 31 Here are some very useful ways of characterizing propositional formulas. Start by constructing a truth table for the formula and look at the column of values obtained. We... 2 answers 32 The following simple undirected graph is referred to as the Peterson graph. Which of the following statements is/are$\text{TRUE}?$The chromatic number of the graph is$...
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Can I have solutions of ELEMENTS OF DISCRETE MATHEMATICS by C.L.Liu Solutions. Pls help
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suggest some good resources for discrete mathematics
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Let n be a positive integer. Show that $\binom{2n}{n + 1} + \binom{2n}{n} = \dfrac{\binom{2n + 2}{n + 1}}{2}.$
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Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p-s=q-r$. Which one of the following is true about ...
Evaluate the question of the following limits. $\lim_{x\rightarrow 1} \frac{x}{(x-1)^{2}}$
Evaluate the question of the following limits. $\lim_{x\rightarrow \infty} \frac{2x^{3}+3x-5}{5x^{3}+1}$