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MA204+MA284 : Discrete Mathematics
Week 6: Advance applications of the PIE
http://www.maths.nuigalway.ie/~niall/MA284
12 and 14 of October, 2016
1 “Stars and bars”
2
3
4
5
6
7
Multisets
Problems with non-negative integer solutions
Inequalities
NNI equations with lower bounds on solutions
Advanced Counting Using PIE
Derangements
Le probléme de rencontres
General formula
Miscellaneous
Repetitions
Permutations with indistinguishable objects
Exercises
See also Sections 1.5 and 1.6 of Levin’s Discrete Mathematics: an open introduction.
Assignment 1
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Assignment 1 is now closed
Your grades are available from the Blackboard Grade Centre.
The average score was 17.3/20 (85%).
...........................................................................
Which problem from Discrete Mathematics Assignment 1 would
you like to study in more depth?
I plan to provide more detailed, interactive notes for you on some of the topics
covered in Discrete Mathematics Assignment 1.
I will also use this material as part of my project
work for a module on Learning Technologies
(CEL263), that I am studying.
I would be grateful if you would use the following is
survey to indicate which material you would like
developed further:
https://goo.gl/forms/mz6HqIYwA1HhXfyS2
Assignment 2!
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ASSIGNMENT 2 is now open!
To access the assignment, go to
https://webwork.nuigalway.ie/webwork2/1516-MA284
Your USERNAME is:
Your PASSWORD is:
There are 20 questions.
You may attempt each one up to 10 times.
This assignment contributes 10% to your final grade for Discrete Mathematics.
Deadline: 5pm, Thursday, 3 November.
For more information, see Blackboard, or
http://www.maths.nuigalway.ie/~niall/MA284
“Stars and bars”
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Last week we had the following question
Every day you give some apples to your lecturers. Today you have 7 apples.
How many ways can you give them to the 4 lecturers you have today?
We learned that
This is the same as finding the number of ways we can arrange 7 apples
(stars), divided into 4 groups, separated by 3 bars.
Any way can be written with 10 symbols (7 stars and 3 bars): we just
� �
ways.
have to choose where to put the 3 bars. This can be done in 10
3
“Stars and bars”
Multisets (5/30)
Definition (Multiset)
A multiset is a set of objects, where each object can appear more than once.
As with an ordinary set, order does not matter.
Examples:
“Stars and bars”
Multisets (5/30)
Definition (Multiset)
A multiset is a set of objects, where each object can appear more than once.
As with an ordinary set, order does not matter.
Examples:
“Stars and bars”
Multisets (5/30)
Definition (Multiset)
A multiset is a set of objects, where each object can appear more than once.
As with an ordinary set, order does not matter.
Examples:
“Stars and bars”
Multisets (6/30)
How many sets of multisets of size 4 can you from the numbers
{1, 2, 3, 4, 5}?
“Stars and bars”
Multisets (7/30)
Example
MA204 Semester 1 Examination 2014/15: Q2(a)
1. In how many ways can one distribute ten e1 coins to four students?
2. In how many ways can one distribute ten e1 coins to four students so that
each student receives at least e1?
“Stars and bars”
Multisets (7/30)
Example
MA204 Semester 1 Examination 2014/15: Q2(a)
1. In how many ways can one distribute ten e1 coins to four students?
2. In how many ways can one distribute ten e1 coins to four students so that
each student receives at least e1?
Problems with non-negative integer solutions
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A non-negative integer problem
How many non-negative integer solutions are there to the problem
x1 + x2 + · · · + xk = n?
This is the same as...
How many ways are there to distribute n identical objects among k individuals.
The answer is
�
n+k −1
k −1
�
=
(n + k − 1)!
n!(k − 1)!
Problems with non-negative integer solutions
Inequalities (9/30)
Example (Part 1)
1. How many non-negative integer solutions are there to x1 + x2 + x3 = 3?
Problems with non-negative integer solutions
Inequalities (10/30)
Example (Part 2)
2. How many non-negative integer solutions are there to x1 + x2 ≤ 3?
Problems with non-negative integer solutions
Inequalities (11/30)
Looking at this example, it seems that
The number of non-negative integer solutions to
x 1 + x2 + x3 + · · · + x k ≤ n
is the same as the number of non-negative integer solutions to
x1 + x2 + x3 + · · · + xk + xk+1 = n.
Why is that?
Problems with non-negative integer solutions
Inequalities (12/30)
The number of NNI1solutions
· + xk ≤ n, is exactly the
1 + x2 +
2 to x3
4 x3 + · · 5
same as the number of NNI solutions to x1 + x2 + x3 + · · · + xk < n + 1.
Example (Q2(a), MA284, Semester 1 Exam, 2014/15)
Find the number of non-negative integer solutions of the inequality
x1 + x2 + x3 + x4 + x5 < 11
Answer: this is same as
x1 + x2+ x 3+ x4 + x5 < 10
which is the same as
x1 + x2+ x 3 + x4 + x5 + x =10
6
Finished here Wednesday