Math325 extra credit problem:
Use inclusion-exclusion principle to solve:
Find the number of five-digit combinations from the set cfw_1, 2, 3, 4, 5 in which:
(a) Some digit appears at least three times.
(b) No digit appears more than twice.
Math325 practice Test 2
(on 6.1 to 6.5, 7.1 to 7.4)
1. (6.1) Let n be a positive integer 6. How many different ways are there of rolling n dice
so that each of the numbers 1, 2, , 6 occurs at least once? (Regard the dice as being
distinguishable from one
Math325 Test 1
(on 2.1 to 2.6, 3.1, 3.2, 5.1, 5.2, 5.4, 5.5)
No calculators! Show your work! Total time: 50 minutes. Total points: 100
1'. , (14 pts) How many even numbers between 100,000 and 900,000 (inclusive) have distinct
digits? Express your answer
Math325 practice Test 1
(on 2.1 to 2.6, 3.1, 3.2, 5.1, 5.2, 5.4, 5.5) i
1. Consider 4-digit numbers Whose digits are either 1, 2, 3, 4, or 5.
a. How many numbers are there?
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' c. How many odd numhere if the digits are distinct?
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Math325 practice Test 1
(on 2.1 to 2.6, 3.1, 3.2, 5.1, 5.2, 5.4, 5.5)
1. Consider 4-digit numbers whose digits are either 1, 2, 3, 4, or 5.
a. How many numbers are there?
b. How many numbers are there if the digits are distinct?
c. How many odd numbers ar
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Math325 Test 2
(on 6.1 was, 7.1 to 7.4)
Name
1. (l6pts) Three married couples are seated together at the counter at Monty’s Blue
Plate Diner, occupying six consecutive seats. How many arrangements are there with
no wif
2.2 Permutation and
Combination
Selection: the creation of a subset of a given set.
The subset formed is called a combination.
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Arrangement: is a selection and an ordering of
some or all of the objects from a set of objects,
so that each selected objec
1.3: Tiling chessboard with dominoes
(combinatorial construction)
Domino: 1 by 2 rectangle
Tiling:
Existence question:
Given a chessboard, can we construct a tiling of the
board With dominoes? If no construction is found,
can it be explained Why no tiling
1.6 Addition and Multiplication Principles
Enumerative combinatorics: counting number of
elements in a set
1.6.1 Addition principle
EX; An entertainment guide recommends 6
restaurants and 3 plays that appeal to a couple. 1f the
couple goes to dinner or a
1.5 Counting tilings of rectangles
. From 1.3, we did tiling for m by n chessboard
and raised an enumerative question: how many
Tilings are there if they exist? Here we’ll
discuss some cases when m=l. w
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1.5.lT111ng a rectangle With squares of t
2.3 Combinatorial Models
Obj ectivc: to use Combinatorial models to prove
identities.
Trick: start With easier side, create a model that
counts size of some set, then explain that the other
side counts the same number.
Tiling models
EX; Recall that the co
1.4 Figurate Numbers
In ancient times, Pythagoreans think
eVerything is in numbers:
Triangular numbers
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Math325 practice Test 2
(on 6.1 to 6.5, 7.1 to 7.4)
1. (6.1) Let n be a positive integer 2 6. How many different ways are there of rolling n dice
so that each of the numbers 1, 2, ., 6 occurs at least once? (R