Discrete Mathematics Answers

a) How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen? b) How many must be selected to guarantee that at least three hearts are selected?

01. A) Let a, b, and c be integers, where a = 0. Then (i) If a | b and a | c, then a | (b + c); (ii) If a | b, then a | bc for all integers c; (iii) If a | b and b | c, then a | c. Course Code: CSE-1102 B) Use Algorithm of Modular Exponentiation to find 1231001 mod 101

Find a counter example to the following universally quantified statements, where the domain for all variables consist of all integers. ∀x, x^2, <=x^3.

Width of a tree is defined as the largest distance(number of edges in the shortest path) between any two vertices. What is the smallest possible width of the tree with 100 vertices?

1


2


99


101
1 point
How many complete graphs with at least 1 edge and at most 50 edges are there?

8


9


10


11

Q.Which of the following can be a degree sequence of a simple graph?

{4,4,3,2,1}


{4,3,2,1}


{4,4,3,2}


{4,2,2,2,2}

Q.If G is a connected simple graph and e,f and g are edges of G, then which of the following is/are true:

There is always a spanning tree of G that contains e


There is always a spanning tree of G that contains e and f


There is always a spanning tree of G that contains e,f and g


All of the above.
Q.If G is a simple, disconnected graph on n vertices, how many maximum edges can it have?

n−2


(n2)


(n−12)


(n−12)+1
Q.Which of the following statements is NOT true?

The maximum number of cut vertices in any simple, connected graph on n vertices is n−2


If there exist a cut-edge in the graph, then a cut-vertex must also exist.


If there exist a cut-vertex in the graph, then a cut-edge must also exist.


The maximum number of cut-edges in any simple graph on n vertices is n−1.

How many bit strings can be formed by using seven 1s and nine 0s?

Compute the number of distinct arrangements of the characters ABBCCCDDDDEEEEE

How many solutions are there to the equation x1+x2+x3+x4+x5+x6= 25 where the xi(for i= 1,2,3,4,5,6) are non-negative integers?

Let S={a, b, c, d, e}. Find the number of ways to select 7 elements from S when repetition is allowed. The order in which the elements are chosen does not matter

Use Mathematical Induction to prove the Binomial Theorem.