Simple answers but work shown thank you!
(a) In how many different ways can the letters of the word
wombat be arranged?
(b) In how many different ways can the letters of the word
wombat be arranged if the letters wo must remain together (in this
order)?
(c) How many different 3-letter words can be formed from the
letters of the word wombat? And what if w must be the first letter
of any such 3-letter word?
(a) What is the maximum degree of a vertex in a simple graph
with n vertices?
(b) What is the maximum number of edges in a simple graph of
n vertices?
(c) Given a natural number n, does there exist a simple
graph with n vertices and the maximum number of edges?
Give an algorithm to find the number of ways you can place knights
on an N by M (M < N) chessboard such that no two knights can
attack each other (there can be any number of knights on the board,
including zero knights). Clear describe your algorithm and prove
its correctness. The runtime should be O(2^3M * N).
a) In how many ways n distinct ball can be given to k children
so that no child gets more than 3 balls?
b) What happens if the balls are indistinguishable?
Let G be a simple undirected graph with n vertices where n is an
even number. Prove that G contains a triangle if it has at least
(n^2 / 4) + 1 edges using mathematical induction.
Q 4 One can think of a circle as an N-sided polygon where N is a
very very large number. This is because as the number of sides of a
polygon increases the shape of the polygon starts resembling that
of a circle. In the light of this fact explain why the diffraction
pattern produced by the aperture DOTS is made of alternating bright
and dark circular bands.
Hint: First determine the diffraction pattern produced by an
N-sided polygon. Then...
Consider an undirected graph G that has n distinct
vertices. Assume n≥3.
How many distinct edges will there be in any circuit for G that
contains all the vertices in G?
What is the maximum degree that any vertex in G can have?
What is the maximum number of distinct edges G can have?
What is the maximum number of distinct edges that G can have if
G is disconnected?